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Measure Less, Know More: Self-Supervised Test-Time Feature Acquisition
Eeshaan Jain, Linus Bleistein, Bart Deplancke, Charlotte Bunne
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Abstract
Abstract:Recent progress in multimodal, high-dimensional learning has enabled foundation models to process heterogeneous, large-scale data. However, at test time, acquiring all features or modalities can be prohibitively costly and often redundant. Sequentially selecting informative modalities is therefore critical, yet challenging when the downstream task or prediction target is unknown. To this end, we introduce ECHO-$k$, a task-agnostic and self-supervised learning principle for modality acquisition: we use a deep model's internal pretrained representations (e.g., from a foundation model) as proxy targets that summarize cross-modal information. We provide theoretical guarantees in a stylized linear setting that motivate a reinforcement learning (RL) policy for sequential modality selection. Across task-agnostic and label-free acquisition baselines, ECHO-$k$ consistently improves budgeted downstream performance across diverse foundation-model backends. Our method provides a principled route to cost-aware test-time deployment, with implications for any multimodal system where measurements are expensive or time-constrained, and downstream tasks unknown a priori.
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- Source: https://arxiv.org/abs/2610.03454v1
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Measure Less, Know More: Self-Supervised Test-Time Feature Acquisition Eeshaan Jain Affiliation: School of Computer and Communication Sciences, EPFL Email: eeshaan.jain@epfl.ch Linus Bleistein Affiliation: School of Computer and Communication Sciences, EPFL Affiliation: School of Life Sciences, EPFL Email: linus.bleistein@epfl.ch Bart Deplancke Affiliation: School of Life Sciences, EPFL Email: bart.deplancke@epfl.ch Charlotte Bunne Affiliation: School of Computer and Communication Sciences, EPFL Affiliation: School of Life Sciences, EPFL Email: charlotte.bunne@epfl.ch Abstract Recent progress in multimodal, high-dimensional learning has enabled foundation models to process heterogeneous, large-scale data. However, at test time, acquiring all features or modalities can be prohibitively costly and often redundant. Sequentially selecting informative modalities is therefore critical, yet challenging when the downstream task or prediction target is unknown. To this end, we introduce echo-k, a task-agnostic and self-supervised learning principle for modality acquisition: we use a deep model’s internal pretrained representations (e.g., from a foundation model) as proxy targets that summarize cross-modal information. We provide theoretical guarantees in a stylized linear setting that motivate a reinforcement learning (RL) policy for sequential modality selection. Across task-agnostic and label-free acquisition baselines, echo-k echo -k consistently improves budgeted downstream performance across diverse foundation-model backends. Our method provides a principled route to cost-aware test-time deployment, with implications for any multimodal system where measurements are expensive or time-constrained, and downstream tasks unknown a priori. The code is available at github.com/bunnelab/echo-k. 1 Introduction Multimodal foundation models (FMs) thrive on more information: the more modalities we can feed them at test time, the more accurate, robust, and context-aware their representations become (Kaplan et al., 2020; Zhai et al., 2022; Mizrahi et al., 2023; Shukor et al., 2025). In practice, however, not every modality is available by default. Each additional assay, sensor, or data source carries a tangible cost in time, money, throughput, or patient burden. This creates a deployment gap: while models can theoretically produce rich representations by exploiting multiview data, acquiring all views for every instance is infeasible. Modalities are stratified by cost, latency, and availability: in clinical diagnostics, routine measurements are often available immediately, while additional molecular or imaging assays can disambiguate uncertain cases but are constrained by turnaround time, cost, or limited sample material. Similar asymmetries arise in scientific discovery, where inexpensive proxies can be collected at scale, but high-fidelity measurements require specialized instrumentation or scarce experimental time. Measurements are therefore not acquired exhaustively, nor independently, but as structured groups of modalities or features corresponding to coherent sensing processes. We refer to each such group as a view. Under a fixed budget, only a small number of views can be queried per instance, and the value of each view depends on which views have already been observed. The acquisition problem is therefore inherently sequential and adaptive: at test time, given the views observed so far, which view should a foundation model acquire next to most improve its downstream predictions? Figure 1: echo-k echo -k learns an RL policy to iteratively selects views to approximate the full latent representation learned by any pretrained model, without any knowledge of downstream tasks. Once learned, the policy can be deployed for budgeted feature acquisition. Existing approaches to sequential acquisition are typically task- and data-driven: they define utility with respect to a specific downstream objective, such as classification or regression, and optimize expected task loss (Gong et al., 2019; Gadgil et al., 2023; Guney et al., 2025). This paradigm requires labeled data at acquisition time, task-specific reward design, or explicit modeling of task uncertainty. As a result, acquisition policies are tightly coupled to a particular task and dataset, and must be retrained or redesigned when downstream objectives change. This reliance on supervision has left the fully task-agnostic, label-free acquisition setting largely unaddressed (see Related Work in § B). Yet modern pretrained models are increasingly deployed as frozen backbones whose representations support many downstream tasks. This shift suggests a different perspective on acquisition: rather than selecting measurements for a single supervised objective, one can ask which measurements are needed to preserve the representation formed by the pretrained model. In this work, we propose to use these latent representations learned by modern foundation models as a task-agnostic signal for view acquisition. When trained on heterogeneous, multiview data, these models learn compressed representations that integrate complementary evidence across views into a compact state (Radford et al., 2021; Siméoni et al., 2025; Wenckstern et al., 2025; Oquab et al., 2023; Lu et al., 2024; Qu et al., 2025; Shmatko et al., 2025). This latent state encodes the information the model deems relevant across many downstream tasks. We use these rich latent states as a learning objective: rather than defining utility with respect to a specific downstream objective, we learn a view selection policy that iteratively selects views to approximate the full latent representation from a minimal number of views. Learning acquisition policies in representation spaces has two main advantages. On the one hand, it removes dependence on downstream supervision: acquisition decisions can be learned without task labels or task-specific losses. On the other hand, it enables broad transfer: because the representation is shared across tasks, preserving it supports many downstream objectives simultaneously. If the partial-view representation closely matches the full-view representation, then any downstream predictor operating on that representation should incur minimal degradation, independent of the task. We instantiate this principle with echo-k, a reinforcement learning algorithm for long-horizon sequential view acquisition that modularly integrates with any self-supervised (foundation) model (Fig. 1). At each step, the RL agent selects which view to acquire next and the reward quantifies progress toward the corresponding full-view representation. Once trained, the policy is deployed at inference time on unobserved views to allocate acquisition budgets on an instance-by-instance basis. We support this algorithm with the first theoretical analysis of self-supervised feature acquisition: in a linear model, where we show that representation recovery controls the latent representation regret (Theorem 3.5), and that conditioning on partial observations yields provable gains over fixed orderings. These results in turn justify the RL reward design and motivate the long-horizon acquisition strategy, enabling cost-effective deployment of large multimodal models in settings where full multiview acquisition is impractical. Contributions. Our contributions are as follows. 1. We introduce a model-driven framework for self-supervised feature acquisition, in which acquisition decisions are guided by the internal representations of pretrained models, without access to task-specific supervision. 2. We provide the first theoretical analysis of self-supervised feature acquisition. In a stylized linear setting, we derive performance bounds that relate optimal test-time policies and oracle selection strategies, formally characterizing the gap induced by limited information that motivate an RL-based acquisition strategy. 3. We develop echo-k, a cost-efficient reinforcement learning algorithm that learns long-horizon acquisition policies using self-supervised latent representations. The policy is trained to maximize progress toward the full-view representation, translating the representation-recovery principle into a practical sequential acquisition method. 4. Empirically, we show that echo-k echo -k learns effective acquisition strategies across diverse data modalities and acquisition horizons. Across image, tabular, single-cell transcriptomics, spatial proteomics, and multimodal geospatial benchmarks, the learned policies rapidly concentrate informative views in early acquisition stages, validating the effectiveness and generality of representation-driven acquisition under realistic budget constraints. Our contributions bridge theory and practice in unsupervised feature acquisition, enabling foundation models to reason about what to measure next under realistic constraints. 2 Representation-Driven Acquisition as a Markov Decision Process In the absence of a specified downstream task, acquisition requires a notion of value that is independent of labels. We define this value through the pretrained multiview encoder: acquired views should recover the representation obtained from all views. Setup. Concretely, we consider n i.i.d. samples, each with K possible views i=(1i,…,Ki) x^i=( m_1^i,…, m_K^i) drawn from a joint distribution over the product space of multiple views ℳ1×⋯×ℳKM^1×·s×M^K. 11 1 Throughout, we use the term view to represent either a modality or a feature group acquired jointly, as our framework applies uniformly to all such settings. After selection, each view k m_k is processed by a view-specific encoder Φk:ℳk→ℝdk _k:M^k ^d_k 22 2 Note, under this framework, multiple views can also share the same joint encoder, or correspond to unimodal encoders. and combined via a fixed aggregation :ℝd1×…ℝdK→ℝpA_W:R^d_1×…R^d_K ^p parameterized by W, yielding the full-view representation ∗=(Φ1(1),…,ΦK(K)) z^*=A_W\! ( _1( m_1),…, _K( m_K) ). This formulation subsumes both multimodal, where each view corresponds to a distinct sensing modality, and unimodal models in which views are partitioned into acquisition units but processed by a shared encoder. For notational simplicity, we absorb encoders into the views and write ∗ z^* directly. We let t⊆[[1,K]] P_t [\![1,K]\!] be the acquired set after t steps, with induced partial representation (t):=(Φk(k):k∈t) z( P_t):=A_W\! (\ _k( m_k):k∈ P_t\ ). Objective. We measure acquisition quality through the representation-recovery loss ℒ(t):=‖(t)−⋆‖22,L( P_t):=\| z( P_t)- z \|_2^2, and seek a sequence (k1,…,kT)(k_1,…,k_T) minimizing ℒ(T)L( P_T) under budget T≪KT K. The ideal one-step oracle greedy rule is kt+1∈argmink∉t‖(t∪k)−∗‖22.k_t+1∈ _k∉ P_t \| z\! ( P_t∪\k\ )- z^* \|_2^2. (1) From training to test-time deployment. The ideal policy defined above is not deployable at test time. Indeed, it requires both the full-view representation ⋆ z and forward passes for every candidate view. During training, we use unlabeled data to construct a self-supervised representation-recovery signal to train a view-selection policy. In the complete-reference setting, fully observed training samples allow us to compute the full-view representation ⋆ z and use it as the reference target. More generally, when only a subset of views Ri⊆[[1,K]]R_i [\![1,K]\!] is available for training instance i, we can use the corresponding reference representation (Ri) z(R_i) as a partially observed target. In both cases, downstream labels are not used to train the acquisition policy. At deployment, the learned policy acts solely from the current acquisition state st=(t,(t),bt)s_t=( P_t, z( P_t),b_t); ⋆ z is never accessed. Downstream tasks. A representation-recovery objective is only meaningful if minimizing it translates to better performance on downstream tasks. If g is an LgL_g-Lipschitz downstream predictor and the loss ℓ is LℓL_ -Lipschitz in the prediction, then for any acquired set P, ℓ(g(()),y)−ℓ(g(∗),y)≤LℓLg‖()−∗‖2. \! (g( z( P)),\,y )- \! (g( z^*),\,y )\;≤\;L_ L_g\,\| z( P)- z^*\|_2. This provides a simple sufficient condition under which latent recovery bounds the degradation of a fixed downstream predictor. In our experiments, downstream labels are not used to train the acquisition policy; they are used only post hoc to evaluate whether the recovered representations preserve task-relevant information. The training-deployment asymmetry shown above raises the central theoretical question: how much is lost when acquisition proceeds from partial observations, relative to an oracle greedy policy that can evaluate the full-information value of each view? A complete MDP formulation is given in § C. 3 Theoretical Guarantees To quantify the cost of this partial observability, we compare two idealized strategies in a stylized linear setting: (i) an oracle policy that observes the full view vector x and greedily selects the view that most reduces representation error, and (i) an optimal test-time policy that makes the same greedy choice but conditions only on previously acquired views. The gap between such oracle and limited-information strategies has been studied in supervised settings (He et al., 2012; Madasu et al., 2022; Valancius et al., 2023); our contribution is to characterize it in the self-supervised, representation-recovery setting, where no task labels are available. Let ∈ℝd x ^d be a random view vector and ∗=():=∈ℝm z^*= z( x):=W x ^m a latent representation where =(ik)∈ℝm×dW=(w_ik) ^m× d is a fixed linear aggregation operator. This setting captures the last-layer behavior of linear or locally linearized representation models. Policies iteratively construct a view subset t⊆[[1,d]] P_t [\![1,d]\!]; at step t, the quality of the selected set t P_t is measured by the conditional expected square reconstruction error ℒ(t∣ℐt) ( P_t _t) :=∑i=1m[(∑k=1dikxk−∑k∈tikxk)2|ℐt]=∑i=1m[(∑k∉tikxk)2|ℐt], := _i=1^mE\! [ ( _k=1^dw_ikx_k- _k∈ P_tw_ikx_k )^2 |\;I_t ]= _i=1^mE\! [ ( _k∉ P_tw_ikx_k )^2 |\;I_t ], where ℐtI_t denotes the information available to the policy at time t, which is different for each policy. One-step oracle policy. The one-step oracle policy greedily selects views with full access to the realization of x. Given a set t⋆ P_t selected at time t, the oracle chooses kt⋆ k_t ∈argmink∉t⋆∑i=1m(∑j∉t⋆∪kijxj)2=argmink∉t⋆∑j∉t∗∪k‖j‖22xj2⏟=:j+∑j≠j′j,j′∉t∗∪kj⊤j′xjxj′⏟=:j′, ∈ _k∉ P_t _i=1^m ( _j∉ P_t ∪\k\w_ijx_j )^2= _k∉ P_t _j∉ P_t^*∪\k\ \|w_j\|_2^2x_j^2_=: E_j+ _ subarraycj≠ j \\ j,j ∉ P_t*∪\k\ subarray w_j w_j x_jx_j _=: C_j , where kw_k denotes the k-th column of W. The oracle therefore exploits both individual view contributions k E_k (energies) and cross-view interaction terms Ckk′C_k , made possible by full knowledge of x. A test-time policy, by contrast, cannot evaluate the Ckk′C_k terms, since they depend on the unobserved views; it must instead rely on conditional expectations given observed views. Test-time policies. We define the class of one-step test-time policies as stochastic mappings that, at each step, select a new view based solely on the history of previously observed views and update their beliefs over the unobserved ones. Formally, such policies are adapted to the filtration ℱt:=σ(xk:k∈t)F_t:=σ(x_k:k∈ P_t) and therefore operate under partial observability, in contrast to the oracle. Among this class, the optimal test-time policy greedily minimizes the conditional expected representation loss, with perfect knowledge of the conditional distribution of unobserved views given observed ones. Given the current selected set ^t P_t, this policy chooses k^t∈argmink∉^t[∑j∉^t∪k[j|ℱt]+∑j≠j′j,j′∉^t∪k[j′|ℱt]], k_t∈ _k∉ P_t [ _j∉ P_t∪\k\E[ E_j|F_t]+ _ subarraycj≠ j \\ j,j ∉ P_t∪\k\ subarrayE[ C_j |F_t] ], where j E_j and jj′ C_j are as defined in the oracle decomposition. To quantify performance, we define the true loss incurred after selecting t P_t as ℒ(t,):=‖∑k∉tkxk‖22L( P_t, x):= \| _k∉ P_tw_kx_k \|_2^2, and the latent regret of the test-time policy as ℛt:=[ℒ(^t,)]−[ℒ(t⋆,)],R_t:=E_ x [L( P_t, x) ]-E_ x [L( P _t, x) ], where t⋆ P_t denotes the view set selected by the oracle after t steps. Proxy loss. The true loss includes pairwise interaction terms jj′ C_j , so the marginal value of a view depends on the entire set of unobserved views; thus bounding expected regret would require controlling the distribution of orderings in a quadratically coupled system, which is analytically intractable without additional structure. To isolate the effect of partial observability from this combinatorial difficulty, we analyze a proxy objective ℒ~(t,):=∑k∉Pt[k∣ℐt] L( P_t, x):= _k∉ P_tE[ E_k _t] and connect it back to the true regret in Theorem 3.5. Two quantities govern the proxy analysis: the centered variables j,s:=j−[j|ℱs] Y_j,s:= E_j-E[ E_j|F_s] measure how far each view’s energy deviates from its conditional expectation at step s, and the maximal conditional deviation s:=maxi,j∉^t(Yi,s−Yj,s) R_s:= _i,j∉ P_t(Y_i,s-Y_j,s) captures the worst-case spread of these estimation errors among unselected views. 3.1 Pathwise Decomposition of the Proxy Regret We now study the gap between the one-step oracle policy and the optimal test-time policy under the proxy objective. Lemma 3.1 provides a pathwise bound on their difference. Lemma 3.1. For all t≥2t≥ 2, the proxy loss gap satisfies ℒ~(^t,)−ℒ~(t⋆,)≤∑s=1tk^s∈Bts, L( P_t, x)- L( P _t, x)≤Σ _s=1^t 1_\ k_s∈ B_t\ R_s, (2) where BtB_t denotes the mismatch set between the oracle and test-time selections up to time t. Lemma 3.1 reduces the proxy regret to a sum of weighted mistake events: regret is incurred only when the test-time policy selects a view outside the oracle-selected set, and the magnitude of each mistake is controlled by the conditional deviation s R_s. Crucially, extending this bound beyond a pathwise statement requires detailed control over the distribution of the order statistics of (k)k=1d( E_k)_k=1^d. In particular, bounding the expected regret entails enumerating all possible orderings of the view energies and characterizing the frequencies and magnitude of deviation of the two policies. Without structural assumptions on the joint distribution of (k)( E_k), we find such a control to be intractable.We therefore posit a latent-variable model in which the data is driven by a shared discrete factor with Gumbel noise, which induces a tractable Plackett–Luce distribution over view rankings (Yellott Jr, 1977; Ma et al., 2021) and makes the regret analysis possible. Assumption 3.2 (Gumbel latent correlation model). The view vector ∈ℝd x ^d is strictly positive and satisfies log=+⊙Z+ε, x= η+ β Z+ , where ,∈ℝd η, β ^d are deterministic, and the common latent factor Z is a discrete latent variable taking values in z1,…,zMz_1,…,z_M with probabilities p1,…,pMp_1,…,p_M. The noise terms follow εj∼i.i.d.Gumbel(0,σ) _j i.i.d. Gumbel(0,σ) with σ∈(0,1/4)σ∈(0,1/4). We further assume that for all latent states zmz_m, ℙ[Z=zm∣ℱs]>0P[Z=z_m _s]>0. Under Assumption 3.2, the conditional ordering of the energies (j)j=1d( E_j)_j=1^d given Z=zmZ=z_m follows a Plackett–Luce distribution. Specifically, for any permutation π of 1,…,d1,…,d, ℙ(π(1)≥⋯≥π(d)∣Z=zm)=∏s=1dehπ(s),zm∑r=sdehπ(r),zm, ( E_π(1)≥…≥ E_π(d) Z=z_m )=Π _s=1^d e^h_π(s),z_mΣ _r=s^de^h_π(r),z_m, where (hj,zm)j,m(h_j,z_m)_j,m are model-dependent utilities determined by (,,σ)( η, β,σ). Marginally, the induced ranking distribution is therefore a mixture of Plackett–Luce models over the latent states of Z. The role of the Gumbel assumption is not primarily to model the data distribution itself, but to ensure that the ranking of view energies follows a tractable parametric family, which is what makes the regret analysis possible. Fig. 5 provides visualizations of datasets generated under this model. 3.2 Regret Bounds under the Gumbel Model For each latent state m, define the utilities uj(m):=2log(‖j‖2)+2ηj+2βjzmu_j^(m):=2 (\|w_j\|_2)+2 _j+2 _jz_m and let γ:=minm∈[M]minj≠j′|uj(m)−uj′(m)|>0γ:= _m∈[M] _j≠ j |u_j^(m)-u_j ^(m) |>0 denote the minimum separation between utilities. A larger γ allows easier identification of view-energy rankings. Also, denote pmin=minm=1,…,Mpmp_ = _m=1,…,Mp_m. A key step is to bound the number of disagreements between the oracle and optimal test-time policies. Let GtG_t (resp. BtB_t) denote the set of views selected only by the oracle (resp. optimal test-time) policy at time t, and define Rmax:=maxj∈[d]maxme2βjzmminme2βjzm<∞, and εγ:=eγ−1Rmax+eγ−1∈(0,1).R_ := _j∈[d] _me^2 _jz_m _me^2 _jz_m<∞, and _γ:= e^γ-1R_ +e^γ-1∈(0,1). Lemma 3.3. The expected number of oracle-test-time mismatches satisfies, for any λ>0λ>0, |Bt|=|Gt|≤Dt(λ):=mint,d−t,Cλ1−ρλ+1eρ∗−1E|B_t|=E|G_t|≤ D_t(λ):= \t,d-t, C_λ1- _λ+ 1e _*-1 \ (3) where Cλ:=(M−1)(1−εγεγpmin)λC_λ:=(M-1) ( 1- _γ _γp_min )^λ, ρ⋆:=1−e−γ/(2σ) _ :=1-e^-γ/(2σ), and ρλ _λ is a contraction constant (see § F.4). Proof outline. The analysis proceeds in three steps: (i) we bound the number of oracle–test-time disagreements (i) this yields a proxy regret bound via the pathwise decomposition; (i) we relate proxy and true losses through a structural assumption on the embedding weights. For proofs, see § E. The regret on the proxy controls errors due to partial observability, but says nothing about the gap between the proxy and true losses. Bridging this gap requires a condition on the geometry of the embedding weights: if the normalized columns of W are approximately orthogonal, the cross-view interaction terms jj′ C_j , that the proxy ignores, remain small. Assumption 3.4 (Near Isometry). There exists constants C, peff>0p_ eff>0 such that for any subset J⊆1,…,dJ \1,…,d\ of size m, the matrix J=(u/‖u‖)u∈J B_J= (w_u/ \|w_u \| )_u∈ J satisfies ‖J⊤J−m‖op≤C(mpeff+mpeff). \| B_J B_J- I_m \|_ op≤ C ( mp_ eff+ mp_ eff ). This condition enforces approximate orthogonality of embedding directions and is empirically satisfied by large-scale vision transformers such as the DINOv2 family (Oquab et al., 2023) and by trained autoencoders across a range of dimensions (§ H.9.2). It ensures that the cross-view interaction terms ignored by the proxy objective remain controlled, so that the proxy loss remains aligned with the true latent reconstruction loss. We can now state the main result. Theorem 3.5. Take p∈(1,12σ)p∈ (1, 12σ ) and let q=p−1q= pp-1. Under Assumptions 3.2 and 3.4, for any λ>0λ>0, the latent regret satisfies ℛt≤ _t≤ (∑i=1dM(i)min(t,d−i+1))1/pDt1/q+C′(Ntpeff+Ntpeff)DtNt, ( _i=1^dM_(i) (t,d-i+1) )^1/pD_t^1/q+C ( N_tp_ eff+ N_tp_ eff ) D_tN_t, (4) where Mj:=[jp]M_j:=E[ E_j^p] and M(1)≥⋯≥M(d)M_(1)≥·s≥ M_(d) be the sorted p-th moments, and Nt=d−t+DtN_t=d-t+D_t. Figure 2: Expected true loss gap curve between the oracle and the test-time optimal, random, and prior-based ([j]E[ E_j]) policies. The bound separates two sources of error. The first term is an oracle-mismatch term: it captures the cost of partial observability through the disagreement envelope DtD_t. This term decreases as the utility gap γ increases, reflecting that well-separated energy rankings are easier to infer from partial observations. The second term is a spectral misalignment term: it controls the error introduced by replacing the true latent loss with the marginal-energy proxy. Consequently, the bound is boundary-correct: since Dt=0D_t=0 at t=0t=0 and t=dt=d, both terms vanish at the endpoints, matching the reverse-U regret profile observed empirically in Fig. 2. 4 Methodology 4.1 From Theoretical Test-Time Policies to Learnable Policies The theoretical analysis in § 3 yields three insights that directly inform algorithm design. First, Theorem 3.5 shows that representation-recovery errors control latent regret up to an oracle-mismatch term and a spectral misalignment term, which becomes small when embedding directions are approximately orthogonal. This motivates using progress toward the full-view representation as a self-supervised acquisition signal. Second, the posterior concentration result (Lemma F.4) shows that conditioning on observed views can reduce the gap to a full-information oracle, motivating adaptive, instance-specific policies that update after each acquisition. Third, the analysis is restricted to myopic policies: the oracle itself is one-step greedy, and the regret bound characterizes the gap between two greedy strategies. In practice, the conditional law of unobserved views given ℱtF_t is unknown, and acquisition is a finite-horizon decision problem in which early choices affect the value of later measurements, especially when views are correlated (Shim et al., 2018; Zannone et al., 2019; Kachuee et al., 2019; Schütz et al., 2025). This motivates moving beyond the greedy regime to long-horizon planning. Together, these insights culminate in echo-k echo -k, a reinforcement-learning algorithm for self-supervised, long-horizon view acquisition. We use the theory to define a reward, and learn a parameterized policy that optimizes cumulative improvement in representation quality over an entire acquisition trajectory, rather than approximating the one-step oracle. Formally, we aim to learn a parameterized policy π π_ θ that maximizes cumulative discounted improvement starting from any step s max∑t=sd∈αt−s[Rt(π(t−1))], _ θ∈ \,\,Σ _t=s^dα^t-sE [R_t ( _ θ( P_t-1) ) ], where RtR_t denotes the incremental improvement in latent representation quality at step t, α∈(0,1]α∈(0,1] is a discount factor that allows the policy to trade off immediate gains against future benefits. The self-supervised reward requires a reference representation against which partial representations are compared. We consider two training regimes. In the complete-reference setting, each training instance is fully observed, so the reference is the full-view representation i⋆=i([[1,K]]) z_i = z_i([\![1,K]\!]). In the partially observed-reference setting, training instance i contains only a subset of available views Ri⊆[[1,K]]R_i [\![1,K]\!]; in this case, we use iref=i(Ri) z_i^ref= z_i(R_i) as a noisy but label-free reference target. 4.2 Learning Representation-Driven Policies Reward. Given the reference representation iref z_i^ref, we define the representation error for instance i as ℓi(S):=‖iref−i(S)‖2 _i(S):=\| z_i^ref- z_i(S)\|_2. The step-wise reward is the normalized improvement in the error, Rt(π^):=ωt1N∑i=1Nℓi(t−1i)−ℓi(t−1i∪kti)ℓi(∅)+ϵ,R_t( π_ θ):= _t 1N _i=1^N _i( P_t-1^i)- _i( P_t-1^i∪\k_t^i\) _i( )+ε, (5) where N is the minibatch size, and kti∼π^(⋅∣st−1i)k_t^i π_ θ(· s_t-1^i) is the view selected for instance i at step t. We use the unsquared norm rather than the squared loss analyzed in § 3, as this provides more stable reward scaling in practice. The reward 5 retains the same structure as the representation-recovery objective studied theoretically: it measures marginal progress toward a reference latent representation without using downstream labels. The factor ωt _t optionally upweights early acquisition steps, where marginal gains are typically largest, and normalization prevents high-error instances from dominating the learning signal. Moreover, when views have nonuniform acquisition costs ckc_k, we use a cost-aware reward of the form rtcost=rt−λc1N∑i=1Ncktir_t^cost=r_t- _c 1N _i=1^Nc_k_t^i with costs normalized across views. Policy input. At each step, the policy receives the current partial representation i(ti) z_i( P_t^i), the observed view values with unobserved entries masked, the acquisition mask, and the remaining budget. In heterogeneous-cost experiments, the state is additionally augmented with the view-cost vector and remaining cost budget; the full state specification is given in § G.5. Optimization. We optimize π π_ θ using Proximal Policy Optimization (PPO) (Schulman et al., 2017), with invalid action masking (Huang and Ontañón, 2020) to prevent re-selection of already acquired views. During both training and deployment, the policy has access only to the frozen representation model and partial observations induced by the current acquisition set. 5 Experimental Results We evaluate echo-k echo -k across controlled synthetic experiments, vision, and tabular encoders, and three frozen foundation-model settings: single-cell transcriptomics with TABULA (Ding et al., 2025), spatial proteomics with VirTues (Wenckstern et al., 2025), and multimodal geospatial sensing with a self-supervised multimodal encoder (Gordon et al., 2026). The experiments are designed to answer four questions: (i) whether representation-driven acquisition improves downstream performance under limited budgets; (i) whether the same label-free acquisition principle composes with heterogeneous pretrained backbones; (i) whether learned acquisition policies transfer to downstream tasks unseen during policy learning; and (iv) whether the method remains effective under realistic deployment constraints, including partially observed reference data and heterogeneous acquisition costs. We discuss the results for foundation models, grouped by application domain, here. Additional controlled benchmarks on synthetic data, and vision datasets such as MNIST, Fashion-MNIST, and MiniBooNE, a physics benchmark, along with incorporating costs and adaptive stopping, are shown in § H. Evaluation. Across all experiments, the representation model is frozen and receives partially observed inputs. echo-k echo -k learns an acquisition policy using only self-supervised rewards derived from progress toward a reference latent representation. In the complete-reference setting, this reference is the full-view embedding; in the partially observed-reference setting, it is computed from the subset of views available for each training instance. Downstream labels are never used for policy learning and are reserved exclusively for post-hoc evaluation. At deployment, the policy observes only the values of the views acquired so far and the acquisition mask, and never accesses the full-view representation or downstream labels. To measure downstream performance, we train a single linear probe per downstream task on partial embeddings using random masking augmentation, and keep it fixed across acquisition budgets. The same probe is applied to the partial embeddings induced by every acquisition method, so that differences in performance reflect only which views were acquired. Moreover, a single acquisition policy is learned per backbone and reused across all budgets and downstream tasks, including tasks that were not specified during policy learning. All results are averaged over three random seeds (§ G.3). Baselines and metrics. We compare against 12 baselines spanning oracle/random references, model-driven heuristics (GIR, ESM2-based orderings), and unsupervised feature selection methods (Laplacian Score, spec-φ3 _3, MCFS, UDFS, SVD-Ent, Variance) adapted as acquisition orders (see § G.4 for details). The Oracle greedily selects the view that maximally reduces the distance to the full-view embedding using unobserved values; it is not deployable at test time and serves as an upper bound on greedy acquisition. Domain-specific baselines are evaluated only where they are defined, e.g., ESM2-based orderings require protein-sequence embeddings and are therefore restricted to protein markers. Since most of these baselines induce a fixed, instance-independent acquisition order, we additionally compare against label-free instance-adaptive policies (§ 5.3) and supervised acquisition methods that use downstream labels (§ H.4). For regression, we report R2R^2 or MSE, and for other real-world tasks, classification accuracy as a function of the acquisition budget t∈[[1,d]]t∈[\![1,d]\!], reported as a fraction of the total number of views. To directly assess the training objective, we further report the normalized representation recovery (§ 5.4). For full evaluation details, refer to § G. Figure 3: a. Results of echo-k echo -k on acquiring geospatial modalities to predict soil pH regression on MMEarth-Bench (d=12d=12) b. echo-k echo -k can operate on single-cell RNA-seq foundation models to sequentially acquire informative genes that support broad set of downstream tasks. We further present comparison of echo-k echo -k to baselines, evaluated on a cell type annotation task on hPancreas (with d=200d=200). 5.1 Evaluation on Foundation Model Backends Single-cell foundation models. In single-cell RNA-seq, selecting the right genes is a critical bottleneck, and the same panel must often support multiple analyses such as cell type annotation, trajectory inference, or gene program discovery that may not be specified at design time (Fig. 3a). We pair echo-k echo -k with TABULA (Ding et al., 2025), a FM for single-cell transcriptomics (§ G.2). Results. We evaluate on hPancreas, a curated single-cell transcriptomics (RNA-seq) collection from human pancreatic tissue (Ding et al., 2025), retaining the top 200 highly variable genes across 14 cell types. We use cell type annotation as a representative downstream task; it is the standard evaluation objective in single-cell analysis and requires the gene subset to preserve fine-grained phenotypic distinctions (§ G.1). echo-k echo -k underperforms slightly during the first ∼ 2–3% of acquisitions but rapidly recovers and consistently outperforms Variance, GIR, and Random thereafter (Fig. 3b). Random performs poorly throughout, reflecting the sparsity of informative genes in high-dimensional expression data. Figure 4: Results of echo-k echo -k on the NSCLC dataset. a. Overview of the experimental setting: given high-dimensional multiplex imaging data, we study how foundation models trained on such data can be leveraged to sequentially acquire informative markers for downstream annotation and discovery tasks. b. Comparison of echo-k echo -k with baseline methods on the cancer subtyping task. c. Comparison of echo-k echo -k with baseline methods on the cancer grading task. d. Scatter plot illustrating the accuracy trade-off between cancer subtyping and grading when using the same set of views. Spatial proteomics foundation models. In spatial proteomics, multiplexed protein imaging characterizes tissue through successive rounds of antibody staining, each revealing a new marker and adding to assay cost, imaging time, and tissue degradation risk. Panel design must therefore support multiple diagnostic objectives simultaneously (e.g., cancer subtyping and grading), with different patients potentially benefiting from different markers (Fig. 4a). We compose echo-k echo -k with the VirTues foundation model (Wenckstern et al., 2025), which embeds multiplex imaging data into tissue-level representations. At deployment, the policy mirrors the iterative staining workflow: selecting which marker to acquire next, updating its partial representation, and using the updated embedding to inform the next selection (§ G.5). Results. We evaluate on a non-small cell lung cancer (NSCLC) patient cohort (Cords et al., 2024) profiled by imaging mass cytometry with 40 protein markers spanning tumor, immune, stromal, and vascular compartments. We consider two clinically relevant downstream tasks: cancer subtyping (2 classes) and cancer grading (3 classes), the latter being substantially more challenging (§ G.1). echo-k echo -k consistently outperforms baselines on both tasks (Figs. 4b,c). The scatter plot in Fig. 4d shows that echo-k echo -k achieves favorable accuracy on both tasks under identical acquisition orders, demonstrating task-agnostic panel design. A qualitative analysis (§ H.10) reveals that echo-k echo -k prioritizes clinically established markers such as panCK, Ki-67, and CD4 early in the process, consistent with domain knowledge (Martin et al., 2004; Su et al., 2006), despite receiving no supervision. Acquisition orders differ substantially across samples, with mean pairwise Kendall’s τ=0.06τ=0.06. Across training seeds, orders for the same sample show greater agreement: mean τ=0.512τ=0.512 and 61.6% top-five overlap, compared with 12.8% under random ordering. Multimodal geospatial foundation models. Earth observation integrates heterogeneous sensing products, from optical and radar imagery to terrain, land-cover, canopy-height, meteorological, and geolocation-derived variables. We evaluate echo-k echo -k on MMEarth-Bench (Gordon et al., 2026), a globally distributed benchmark of 12 aligned geospatial modalities, treating each modality as a separately acquirable view. Results. We evaluate on soil pH regression as an unseen downstream task, reporting R2R^2 as a function of the number of modalities. echo-k echo -k outperforms non-oracle baselines consistently under limited modality budgets, reaching strong predictive performance after acquiring only 20-30% of the views (Fig. 3b). We also notice that GIR and Variance-based baselines are the second-best performers. 5.2 Robustness to Partially Observed Reference Data In Table 1, we evaluate whether echo-k echo -k remains effective when trained from incomplete views by randomly masking a fraction m of views before computing the reference embedding, and train the policy against this partially observed target. For single-cell transcriptomics FMs, performance is nearly unchanged for m=m=10%, remains strong for m=m=20%, and still recovers high accuracy under heavier missingness once moderate acquisition budgets are allowed. For proteomics FMs, performance degrades more gradually with m, but partially observed-reference policies remain competitive with strong baselines across budgets. We also notice that for high budgets, GIR is competitive with echo-k echo -k. Overall, echo-k echo -k degrades smoothly rather than collapsing, suggesting that useful acquisition policies can be learned from incomplete reference representations. Table 1: Results of robustness to partially observed reference data. Performance of echo-k echo -k when an m-fraction of views is randomly masked while computing the training reference representation. Accuracy is reported at different acquisition budgets on hPancreas and NSCLC grading. Lower m corresponds to more complete reference data. denotes best and second best performance, apart from the Oracle. Method hPancreas Method NSCLC (Grading) 20% 40% 60% 80% 20% 40% 60% 80% Oracle 0.872±0.0020.872 ± 0.002 0.872±0.0010.872 ± 0.001 0.872±0.0010.872 ± 0.001 0.872±0.0010.872 ± 0.001 Oracle 0.665±0.0060.665 ± 0.006 0.672±0.0100.672 ± 0.010 0.680±0.0120.680 ± 0.012 0.681±0.0100.681 ± 0.010 Var 0.679±0.0040.679 ± 0.004 0.823±0.0040.823 ± 0.004 0.873±0.0010.873 ± 0.001 0.874±0.0010.874 ± 0.001 ESM-Max 0.564±0.0050.564 ± 0.005 0.649±0.0040.649 ± 0.004 0.655±0.0070.655 ± 0.007 0.676±0.0080.676 ± 0.008 SPEC-φ3 _3 0.476±0.0090.476 ± 0.009 0.421±0.0120.421 ± 0.012 0.696±0.0010.696 ± 0.001 0.819±0.0040.819 ± 0.004 ESM-Min 0.582±0.0050.582 ± 0.005 0.569±0.0080.569 ± 0.008 0.597±0.0150.597 ± 0.015 0.684±0.0090.684 ± 0.009 MCFS 0.694±0.0000.694 ± 0.000 0.762±0.0040.762 ± 0.004 0.840±0.0060.840 ± 0.006 0.813±0.0090.813 ± 0.009 GIR 0.607±0.0040.607 ± 0.004 0.636±0.0190.636 ± 0.019 0.678±0.0130.678 ± 0.013 0.683±0.0090.683 ± 0.009 Random 0.500±0.0070.500 ± 0.007 0.516±0.0100.516 ± 0.010 0.566±0.0050.566 ± 0.005 0.608±0.0130.608 ± 0.013 Random 0.573±0.0120.573 ± 0.012 0.611±0.0080.611 ± 0.008 0.640±0.0110.640 ± 0.011 0.661±0.0050.661 ± 0.005 echo-k echo -k 0.808±0.0030.808 ± 0.003 0.880±0.0010.880 ± 0.001 0.884±0.0020.884 ± 0.002 0.884±0.0020.884 ± 0.002 echo-k echo -k 0.658±0.0060.658 ± 0.006 0.671±0.0090.671 ± 0.009 0.674±0.0090.674 ± 0.009 0.683±0.0160.683 ± 0.016 echo-k echo -k (m=m=10%) 0.812±0.0030.812 ± 0.003 0.880±0.0020.880 ± 0.002 0.884±0.0030.884 ± 0.003 0.884±0.0020.884 ± 0.002 echo-k echo -k (m=m=10%) 0.655±0.0150.655 ± 0.015 0.671±0.0140.671 ± 0.014 0.669±0.0130.669 ± 0.013 0.678±0.0120.678 ± 0.012 echo-k echo -k (m=m=20%) 0.808±0.0030.808 ± 0.003 0.871±0.002 pink!400.871 ± 0.002 0.883±0.0020.883 ± 0.002 0.883±0.0020.883 ± 0.002 echo-k echo -k (m=m=20%) 0.641±0.0090.641 ± 0.009 0.658±0.0090.658 ± 0.009 0.660±0.0100.660 ± 0.010 0.672±0.0100.672 ± 0.010 echo-k echo -k (m=m=30%) 0.783±0.0040.783 ± 0.004 0.865±0.0040.865 ± 0.004 0.882±0.0020.882 ± 0.002 0.882±0.0020.882 ± 0.002 echo-k echo -k (m=m=30%) 0.615±0.0070.615 ± 0.007 0.653±0.0110.653 ± 0.011 0.656±0.0080.656 ± 0.008 0.671±0.0180.671 ± 0.018 echo-k echo -k (m=m=40%) 0.751±0.0050.751 ± 0.005 0.851±0.0060.851 ± 0.006 0.864±0.0040.864 ± 0.004 0.871±0.0030.871 ± 0.003 echo-k echo -k (m=m=40%) 0.609±0.0100.609 ± 0.010 0.636±0.0110.636 ± 0.011 0.640±0.0070.640 ± 0.007 0.664±0.0090.664 ± 0.009 echo-k echo -k (m=m=50%) 0.733±0.0070.733 ± 0.007 0.832±0.0050.832 ± 0.005 0.861±0.0040.861 ± 0.004 0.872±0.0020.872 ± 0.002 echo-k echo -k (m=m=50%) 0.595±0.0120.595 ± 0.012 0.622±0.0080.622 ± 0.008 0.636±0.0070.636 ± 0.007 0.669±0.0150.669 ± 0.015 5.3 Comparison with Adaptive Acquisition Table 2: Comparison with unsupervised adaptive acquisition on MiniBooNE. Accuracy (%) at different acquisition budgets. denotes best and second best performance. Method 20% 40% 60% CIG 79.8079.80 84.8784.87 86.1786.17 ALG 77.5077.50 83.6083.60 86.1286.12 GSMRL-AIR 79.1879.18 84.7884.78 86.4086.40 echo-k echo -k 81.7081.70 86.8786.87 87.6087.60 Beyond fixed acquisition orderings, we compare echo-k echo -k with label-free, instance-adaptive policies under the same information setting: GSMRL-AIR (Li and Oliva, 2021), based on conditional generative imputations; Adaptive Latent Gain (ALG), which predicts one-step representation gain; and Cluster Information Gain (CIG), which estimates view-wise information gain under a Gaussian mixture model. echo-k echo -k outperforms all three across every budget on MiniBooNE (Table 2) and hPancreas (§H.3). Notably, ALG shares echo-k echo -k ’s state and reward but optimizes only one-step gain, highlighting the benefit of long-horizon acquisition. Finally, echo-k echo -k remains competitive with supervised acquisition policies that have access to downstream labels (§ H.4), quantifying the cost of task agnosticism. 5.4 Comparison of representation-recovery Table 3: Results of representation recovery. Comparison of ρt _t of echo-k echo -k and Variance at different budgets. Higher ρt _t denotes better recovery. denotes best performance. Dataset Method 20% 40% 60% 80% hPancreas Var 0.380.38 0.530.53 0.650.65 0.840.84 echo-k echo -k 0.410.41 0.580.58 0.700.70 0.890.89 MiniBooNE Var 0.350.35 0.540.54 0.600.60 0.730.73 echo-k echo -k 0.410.41 0.600.60 0.710.71 0.800.80 While our main results focus on downstream performance, we also directly evaluate the representation-recovery objective optimized by echo-k echo -k (Eq. 5). For each instance i, we define normalized recovery after t acquisitions as ρi,t:=1−‖i(ti)−i⋆‖2‖i(∅)−i⋆‖2, _i,t:=1- \| z_i( P_t^i)- z_i \|_2\| z_i( )- z_i \|_2, We report ρt _t averaged over test instances. Table 3 shows that echo-k echo -k achieves higher recovery than Variance, the strongest competing baseline on both hPancreas and MiniBooNE, across all budgets. Moreover, recovery and downstream accuracy exhibit the same ordering (Fig. 3b; Fig. 7): better preservation of the full-view representation corresponds to better downstream performance. This provides empirical support for the connection established in Theorem 3.5 between representation recovery and downstream error. 6 Conclusion We presented echo-k, a task-agnostic and label-free framework for sequential test-time view acquisition that uses a pretrained model’s latent representation as a self-supervised proxy: under a cost budget, acquire views so that the induced partial representation rapidly approaches the full-view representation. We support this formulation with a theoretical analysis in a linear setting that characterizes the gap between oracle and feasible test-time policies, and we introduce a practical cost-aware RL algorithm that learns long-horizon acquisition strategies. Experiments with controlled synthetic data and multimodal foundation models across single-cell transcriptomics, spatial proteomics, and geospatial sensing show improved budgeted acquisition, provide evidence for transfer to downstream tasks unseen during policy learning, robustness to partially observed reference data, and extensibility to heterogeneous acquisition costs. The broader implication is that foundation models need not remain passive predictors over pre-collected inputs; they can become resource-aware systems that actively shape the evidence available for inference. 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Instead of requiring downstream labels to decide which view to acquire next, the method uses the model’s own full-view representation as a self-supervised reference. This addresses a practical deployment bottleneck: multimodal and scientific foundation models often benefit from additional inputs, but acquiring all modalities can be infeasible due to cost, latency, limited tissue or sample material, scarce instrumentation time, or experimental constraints. In scientific settings, echo-k could help prioritize which measurements, assays, markers, sensors, or experiments to run next under limited resources. In clinical settings, it could support budget-aware acquisition of follow-up tests or multimodal patient measurements, while keeping downstream decision-making within validated and expert-supervised workflows. The main limitations concern deployment specificity rather than the general formulation. Acquisition costs and constraints may vary across domains, so practical use requires defining the relevant budget, availability, and cost model for the target setting. The method also benefits from reference representations that are informative enough to guide policy learning; our robustness experiments suggest that these references need not always be complete, but the appropriate level of missingness may depend on the application and backbone. Finally, as with any method intended for scientific or clinical use, policies should be evaluated on the intended endpoint and deployment distribution before adoption. Moreover, echo-k echo -k preserves whatever information the pretrained backbone encodes, and does not assert that the full-view representation ⋆ z is optimal for every downstream task; any task-irrelevant information in ⋆ z is therefore inherited from the backbone. Our experiments evaluate transfer across downstream objectives, but do not test explicit distribution shift between the reference-construction data and the deployment domain. Similarly, the full-view representation is not always the best training target: masked references outperform it on MNIST-14 at 20% and 40% budgets and on hPancreas at 20% (§ H), suggesting that masking can act as a mild regularizer during training. Our robustness sweep considers randomly masked reference views and does not identify where partial references fail, since even at 50% masking echo-k echo -k remains competitive with strong baselines. A natural extension is structured incompleteness, where an entire assay or modality is missing for a subset of samples, and understanding how the reference representation interacts with that structure is an interesting direction for future work. These considerations leave substantial room for future work on richer cost models, uncertainty-aware acquisition, and prospective validation in real-world acquisition workflows. Appendix B Related Work Classical feature selection methods aim to find a subset of relevant features that maximizes model performance while excluding irrelevancies (Kohavi and John, 1997; Guyon and Elisseeff, 2003). Early surveys (Blum and Langley, 1997) reviewed both empirical advances and theoretical definitions of feature relevance, formalizing the notions of strong and weak relevance. Several works cast the problem as submodular optimization (Killamsetty et al., 2021; Jain et al., 2023; Wei et al., 2014; Coleman et al., 2019), though their end goal is to improve performance under limited budget during training rather than at test time. Previous work has also centered around greedy heuristics with theoretical guarantees for sequential search (Golovin and Krause, 2011). Another line of work focuses on instance-specific or personalized feature selection (Melville et al., 2004; Li et al., 2018). Beyond selecting a subset of features, a related field is feature importance quantification: given a fixed dataset and (most commonly) a target, the goal is to isolate features that contribute significantly to predictions (Guyon and Elisseeff, 2003; Settles, 2009; Li et al., 2017; Lundberg and Lee, 2017; Sristi et al., 2023; Jain et al., 2023; Lifshitz et al., 2025). Model-specific methods such as (Breiman, 2001) leverage the structure of the model to assign importance ranks, while model-agnostic methods (Fisher et al., 2019) rely on importance metrics applicable to any model. Active learning (AL) incorporates a sample selection policy in the training loop in an online fashion: at each time t, a new sample is queried to augment the existing dataset (Settles, 2009; Zhan et al., 2022; Tharwat and Schenck, 2023; Li et al., 2025). AL methods are particularly well suited for settings where acquiring new samples is expensive. They can be broadly split into three types of strategies (Zhan et al., 2022). Uncertainty-based querying strategies select points based on the estimated information they provide to the learning algorithm, including maximum entropy selection (Shannon, 1948), confidence-based selection (Wang and Shang, 2014), and adversarial methods (Ducoffe and Precioso, 2018). Diversity-based strategies aim to increase the coverage of samples in the dataset, for instance through core-set approaches that build subsets approximating the full dataset (Sener and Savarese, 2017; Claici et al., 2018; Feldman, 2019; Kim and Shin, 2022). Lastly, combined strategies mix both approaches (Yin et al., 2017; Zhdanov, 2019; Shui et al., 2020). Feature acquisition (FA). Our method focuses on designing instance-specific feature acquisition strategies rather than sample acquisition policies (Settles, 2009; Kirsch et al., 2019; Holzmüller et al., 2023; Kolossov et al., 2023; Gupte et al., 2024). FA strategies can be broadly split (Rahbar et al., 2025) into greedy strategies that sequentially acquire information by maximizing expected utility, and non-greedy methods that plan acquisition over multiple time steps (Zubek and Dietterich, 2002; Dulac-Arnold et al., 2011; Shim et al., 2018; Huang et al., 2018; Huang et al., 2025; Norcliffe et al., 2025; Piskorz et al., 2025), often using reinforcement learning (Shim et al., 2018; Zannone et al., 2019; Kachuee et al., 2019). This problem has also been studied under the name of Bayesian experiment design (BED), which aims to optimize the distribution of ξ, the random variable controlling the experiment, in order to maximize the information gain obtained by running the experiment parametrized by ξ (Chaloner and Verdinelli, 1995; Sebastiani and Wynn, 2000; Cavagnaro et al., 2010; Ryan et al., 2016; Rainforth et al., 2024). Classical BED methods sequentially update a model from a stream of incoming experiments; recently, the focus has shifted towards policy learning (Foster et al., 2021; Blau et al., 2022; Blau et al., 2023) and decision-aware experiment design (Huang et al., 2024). Task-agnostic acquisition. Contrarily to previous work, our proposed method is fundamentally task-agnostic. While some active learning and feature selection strategies have considered weakly or unlabeled settings (Ash et al., 2019; Yehuda et al., 2022; Lifshitz et al., 2025; Kobayashi et al., 2025), task-agnostic feature acquisition has received little attention. Closest to our setup are Li and Oliva (2021), who consider a partially supervised setup and tackle this problem by training a surrogate reward model, and introduce an active instance recognition task in which the goal is to select features to reduce uncertainty. Theoretical aspects of FA have received little attention, both in the supervised and unsupervised setting. Chen et al. (2015) study a probabilistic model inspired by Bayesian experiment design, in which the goal is to learn a policy that sequentially acquires features to infer the value of a target. Kobayashi et al. (2025) and Covert et al. (2023) provide theoretical insights in the supervised case. Our work is the first to consider the unsupervised case and to provide a bound on the gap between the greedy oracle and the optimal test-time policy. Appendix C Extended Mathematical Framework Data generation. We consider n i.i.d. instances. For each i∈[[1,n]]i∈[\![1,n]\!], the data i=(1i,…,Ki) x^i=( m_1^i,…, m_K^i) is drawn from a joint distribution over ℳ1×⋯×ℳKM^1×·s×M^K. We use view, modality, and feature group interchangeably, as our framework applies uniformly to all settings. Representation model. Each view k m_k is processed by a view-specific encoder Φk:ℳk→ℝdk _k:M^k ^d_k. The encoded views are combined through a fixed aggregation function :ℝd1×⋯×ℝdK→ℝpA_W:R^d_1×…×R^d_K ^p, yielding the full-view representation ∗=(Φ1(1),…,ΦK(K)) z^*=A_W\! ( _1( m_1),…, _K( m_K) ). This subsumes both multimodal models (each k m_k is a distinct modality: image, audio, protein marker, …) and unimodal models (features partitioned into jointly acquired groups processed by a shared encoder). For notational simplicity, we drop the encoders in subsequent expressions and treat views as directly mapped into the latent space. Sequential acquisition. Let t⊆[[1,K]] P_t [\![1,K]\!] denote the acquired set after t steps and (t):=(Φk(k):k∈t) z( P_t):=A_W\! (\ _k( m_k):k∈ P_t\ ) the induced partial representation, with unobserved views treated as missing. The finite-horizon acquisition problem is to select a sequence (k1,…,kT)(k_1,…,k_T), with kt+1∉tk_t+1∉ P_t, minimizing the representation-recovery loss ℒ(t):=‖(t)−⋆‖22.L( P_t):=\| z( P_t)- z \|_2^2. Equivalently, the policy maximizes cumulative progress toward ⋆ z along the acquisition trajectory. The ideal one-step greedy rule is kt+1∈argmink∉t‖(t∪k)−∗‖22,k_t+1∈ _k∉ P_t \| z\! ( P_t∪\k\ )- z^* \|_2^2, (6) which is not deployable at test time, since it requires access to both ⋆ z and the representation after acquiring each candidate view. Markov decision process formulation. We frame sequential acquisition as an instance-specific MDP with a continuous state space and evolving action set. Dropping the instance superscript for clarity, the components are: • State. At step t, the state is st=(Mt,t,(t),bt)s_t=(M_t, x_ P_t, z( P_t),b_t), where Mt∈0,1KM_t∈\0,1\^K is the acquisition mask, t x_ P_t are the observed view values, (t) z( P_t) is the current partial representation, and btb_t is the remaining budget. To maintain a fixed state space across steps, unobserved views are set to NA, so the augmented space is (ℳ1∪NA)×⋯×(ℳK∪NA) (M^1∪\ NA\ )×·s× (M^K∪\ NA\ ) (Norcliffe et al., 2025). • Actions. The available actions at step t are the unacquired views: (st)=k∈[[1,K]]:k∉tA(s_t)=\k∈[\![1,K]\!]:k∉ P_t\. • Transitions. Upon selecting kt+1k_t+1, its value is drawn from kt+1∼pdata(⋅∣k,k∈t) m_k_t+1 p_ data(· m_k,k∈ P_t), the state updates to t+1=t∪kt+1 P_t+1= P_t∪\k_t+1\, and (t+1) z( P_t+1) is recomputed. The transition probability between states differing only in coordinate ktk_t is ℙ[t∣t−1]=ℙ[at=kt∣t−1]⋅pdata(kt∣k,k∈t−1).P\! [ P_t P_t-1 ]=P\! [a_t=k_t P_t-1 ]· p_ data\! ( m_k_t m_k,\,k∈ P_t-1 ). • Reward. The reward at step t+1t+1 quantifies progress toward the reference-view representation: rt+1(t+1):=‖(t)−ref‖22−‖(t+1)−ref‖22.r_t+1( P_t+1):=\| z( P_t)- z^ref\|_2^2-\| z( P_t+1)- z^ref\|_2^2. where ref z^ref can be the full view embedding ∗ z^* or the embedding from all the views (can be a subset of full views) measured for the underlying data point. Selection policies. A stochastic selection policy is a map πθ(⋅∣st)∈Δ((st)) _θ(· s_t)∈ (A(s_t)), where Δ((st)) (A(s_t)) is the probability simplex over available actions. During training, fully observed samples allow ⋆ z to be computed and used as a self-supervised target — no downstream labels are required. When only a reference subset RiR_i is observed for instance i, (Ri) z(R_i) serves as a partial target. At deployment, the learned policy is fixed and acts solely from the current state sts_t. Connection to sequential decision making. This MDP is an instance of a K-armed bandit with correlated arms and a single-pull constraint per arm (Gupta et al., 2021); see also Guha and Munagala (2007); Chakrabarti et al. (2008); Carrara et al. (2019) for related setups. The sequential decision problem is Markovian with respect to (t,(t))( P_t, z( P_t)): given the current partial representation and acquired set, future acquisition values are independent of the full history of selections. Appendix D Notation Notation Meaning Representation-acquisition setup n Number of i.i.d. samples. K Number of possible views in the general acquisition problem. ki m_k^i The k-th view of sample i. A view may be a modality or a jointly acquired feature group. ℳkM^k Space of possible values for view k. Φk:ℳk→ℝdk _k:M^k ^d_k View-specific encoder for view k. A_W Fixed aggregation map combining encoded views into a representation. ⋆ z Full-view representation, obtained from all available views. t⊆1,…,K P_t \1,…,K\ Set of acquired views after t acquisition steps in the general MDP formulation. (t) z( P_t) Partial representation induced by the acquired set t P_t. T Acquisition budget. st=(t,(t),bt)s_t=( P_t, z( P_t),b_t) Deployment state of the acquisition policy. btb_t Remaining or current budget component in the MDP state. ℒ(t)L( P_t) Representation-recovery loss, ℒ(t)=‖(t)−⋆‖22L( P_t)=\| z( P_t)- z \|_2^2. Stylized linear model d Number of views/features in the stylized linear theory. m Dimension of the latent linear representation. ∈ℝd x ^d Random view vector in the linear theory. xjx_j The j-th component of the view vector x. ∈ℝm×dW ^m× d Fixed linear aggregation matrix. jw_j The j-th column of W. ∗= z^*=W x Latent representation in the linear theory. ℐtI_t Information available to a policy at time t. This differs between oracle and test-time policies. ℱt=σ(xk:k∈t)F_t=σ(x_k:k∈ P_t) Filtration generated by the views observed up to time t. ℒ(t∣ℐt)L( P_t _t) Conditional expected squared reconstruction error of selected set t P_t given information ℐtI_t. Policies and selected sets t⋆ P_t Set selected by the oracle policy after t steps. ^t P_t Set selected by the optimal test-time policy after t steps. kt⋆k_t View selected by the oracle policy at step t. k^t k_t View selected by the optimal test-time policy at step t. ^−s P_-s Set of views still available to the test-time policy at step s. Losses, energies, and regrets j E_j Energy of view j, defined by j:=‖j‖22xj2 E_j:=\|w_j\|_2^2x_j^2. jj′ C_j Cross-view interaction term, defined by jj′:=j⊤j′xjxj′ C_j :=w_j w_j x_jx_j . ℒ(t,)L( P_t, x) True latent reconstruction loss of selected set t P_t on realization x. ℒ~(t,) L( P_t, x) Proxy loss based only on marginal energy terms. ℛtR_t True latent regret of the optimal test-time policy relative to the oracle policy. ℛ~t R_t Proxy regret of the optimal test-time policy relative to the oracle policy. j,s Y_j,s Centered energy error at step s, j,s:=j−[j∣ℱs] Y_j,s:= E_j-E[ E_j _s]. s R_s Maximal conditional deviation spread among available views at step s. Mj(p)M_j^(p) p-th moment of the energy of view j, Mj(p):=[jp]M_j^(p):=E[ E_j^p]. M(1)(p)≥⋯≥M(d)(p)M_(1)^(p)≥·s≥ M_(d)^(p) Sorted p-th energy moments. AtA_t Moment envelope, At:=∑i=1dM(i)(p)min(t,d−i+1)A_t:= _i=1^dM_(i)^(p) (t,d-i+1). p,qp,q Hölder conjugate exponents, with p∈(1,1/(2σ))p∈(1,1/(2σ)) and q=p/(p−1)q=p/(p-1). Mismatch and ranking events GtG_t Views selected only by the oracle policy at time t. BtB_t Views selected only by the optimal test-time policy at time t. rkA(k) rk_A(k) Rank of k E_k among the energy values (j)j∈A( E_j)_j∈ A. Bad(s,t) Bad(s,t) Bad ranking event, Bad(s,t):=rk^−s(k^s+1)>t−s Bad(s,t):=\ rk_ P_-s( k_s+1)>t-s\. Dt(λ)D_t(λ) Disagreement envelope controlling the expected number of oracle–test-time mismatches. NtN_t Effective size term in the true regret bound, Nt:=d−t+DtN_t:=d-t+D_t. Gumbel latent correlation model Z Discrete latent factor governing correlations between views. z1,…,zMz_1,…,z_M Possible latent states of Z. pmp_m Prior probability of latent state zmz_m. pminp_ Minimum latent-state probability, pmin:=minm∈[M]pmp_ := _m∈[M]p_m. ,∈ℝd η, β ^d Deterministic location and latent-loading vectors in the Gumbel model. εj _j Independent Gumbel noise term for view j. σ Gumbel scale parameter. aja_j Squared column norm, aj:=‖j‖22a_j:=\|w_j\|_2^2, when writing utilities in the form logaj+2ηj+2βjzm a_j+2 _j+2 _jz_m. uj(m)u_j^(m) Utility of view j under latent state zmz_m, uj(m):=logaj+2ηj+2βjzmu_j^(m):= a_j+2 _j+2 _jz_m. θj(m) _j^(m) Plackett–Luce weight, θj(m):=exp(uj(m)/(2σ)) _j^(m):= (u_j^(m)/(2σ)). γ Minimum utility separation, γ:=minm∈[M]minj≠j′|uj(m)−uj′(m)|γ:= _m∈[M] _j≠ j |u_j^(m)-u_j ^(m)|. RmaxR_ Maximum latent-state variation in view weights, Rmax:=maxj∈[d]maxme2βjzmminme2βjzmR_ := _j∈[d] _me^2 _jz_m _me^2 _jz_m. εγ _γ Posterior accuracy threshold, εγ:=eγ−1Rmax+eγ−1 _γ:= e^γ-1R_ +e^γ-1. ρPL(k) _PL(k) Minimum Plackett–Luce selection probability over subsets of size k. ρ⋆ _ Ranking-concentration parameter, ρ⋆:=1−e−γ/(2σ) _ :=1-e^-γ/(2σ). αm(ℱs) _m(F_s) Posterior probability of latent state zmz_m after observing ℱsF_s, αm(ℱs):=ℙ[Z=zm∣ℱs] _m(F_s):=P[Z=z_m _s]. Epost(s,zm)E_post(s,z_m) Posterior failure event under true state zmz_m. cj(m,m′,λ)c_j(m,m ;λ) Chernoff coefficient between the conditional distributions of view j under states m and m′m . ρλ _λ Worst-case Chernoff contraction coefficient, ρλ:=maxj∈[d]maxm≠m′cj(m,m′,λ) _λ:= _j∈[d] _m≠ m c_j(m,m ;λ). CλC_λ Posterior concentration constant, Cλ:=(M−1)(1−εγεγpmin)λC_λ:=(M-1) ( 1- _γ _γp_ )^λ. Spectral misalignment quantities ℳ()M( P) Difference between true and proxy losses for selected set P, ℳ():=ℒ()−ℒ~()M( P):=L( P)- L( P). u()u( P) Vector of unobserved weighted view magnitudes, u():=[j∉∥j∥xj]j=1du( P):=[1_\j∉ P\\|w_j\|x_j]_j=1^d. (,′) U( P, P ) Set of indices not jointly selected by P and ′ P , (,′):=(∩′)c U( P, P ):=( P∩ P )^c. Γ(,′) ( P, P ) Symmetric difference between two selected sets, Γ(,′):=Δ′ ( P, P ):= P P . J B_J Matrix of normalized embedding columns indexed by J, J:=(u/‖u‖)u∈J B_J:=(w_u/\|w_u\|)_u∈ J. peffp_eff Effective dimension parameter in the near-isometry assumption. C,C′C,C Constants appearing in the near-isometry and spectral misalignment bounds. B Matrix of all normalized embedding columns, B:=[j/‖j‖]j=1dB:=[w_j/\|w_j\|]_j=1^d. H Gram deviation matrix, H:=B⊤B−IH:=B B-I. HUH_U Restriction of the Gram deviation matrix to the index set (,′) U( P, P ). Appendix E Proofs E.1 Proof of Lemma 3.1 Proof. Let GtG_t be the set of features selected only by the oracle policy at time t, and BtB_t the set of features selected only by the optimal test-time policy. First, observe that we have the trivial bound |Bt|=|Gt|≤min(d−t,t) |B_t|=|G_t|≤ (d-t,t) We have ℒ~(^t,)−ℒ~(t⋆,)=∑g∈Gtg−∑b∈Btb. L( P_t, x)- L( P _t, x)=Σ _g∈ G_t E_g-Σ _b∈ B_t E_b. Write k^2,…,k^t k_2,…, k_t for the indices of the features selected by the optimal test-time policy. Since GtG_t and BtB_t are of equal size, there exists a permutation σ between BtB_t and GtG_t. Also, by definition of GtG_t, we know that Gt∩Bt=∅G_t∩ B_t=\ \. By iterating over time steps, we have ∑g∈Gtg−∑b∈Btb≤∑s=2tk^s∈Bt(σ(k^s)−k^s). Σ _g∈ G_t E_g-Σ _b∈ B_t E_b≤Σ _s=2^t 1_\ k_s∈ B_t\ ( E_σ( k_s)- E_ k_s ). Now, we also have ∑s=2tk^s∈Bt(σ(k^s)−k^s) Σ _s=2^t 1_\ k_s∈ B_t\ ( E_σ( k_s)- E_ k_s ) = = ∑s=2tk^s∈Bt(σ(k^s)−[σ(k^s)∣ℱs]+[σ(k^s)∣ℱs]−k^s−[k^s∣ℱs]+[k^s∣ℱs]) Σ _s=2^t 1_\ k_s∈ B_t\ ( E_σ( k_s)-E [ E_σ( k_s) _s ]+E [ E_σ( k_s) _s ]- E_ k_s-E [ E_ k_s _s ]+E [ E_ k_s _s ] ) = = ∑s=2tk^s∈Bt(Yσ(k^s),s−Yk^s,s+[σ(k^s)∣ℱs]−[k^s∣ℱs]⏟≤0) Σ _s=2^t 1_\ k_s∈ B_t\ (Y_σ( k_s),s-Y_ k_s,s+ E [ E_σ( k_s) _s ]-E [ E_ k_s _s ]_≤ 0 ) ≤ ≤ ∑s=2tk^s∈Bt(Yσ(k^s),s−Yk^s,s). Σ _s=2^t 1_\ k_s∈ B_t\ (Y_σ( k_s),s-Y_ k_s,s ). where the last equality comes from optimality of the test time policy and the fact that σ(k^s)σ( k_s) is available, i.e. σ(k^s)∈^−sσ( k_s)∈ P_-s . By definition of s R_s, we have ℒ~(^t,)−ℒ~(t⋆,)≤∑s=2tk^s∈Bts. L( P_t, x)- L( P _t, x)≤Σ _s=2^t 1_\ k_s∈ B_t\ R_s. E.2 Proof of Proposition C.1 We prove the following Proposition. Proposition E.1. Take p∈(1,12σ)p∈ (1, 12σ ) and let q=p−1q= pp-1. Under Assumptions, we have ℛ~t≤At1/p(mint,d−t,Cλ1−ρλ+1eρ0−1)1/q R_t≤ A_t^1/p ( \t,d-t, C_λ1- _λ+ 1e _0-1 \ )^1/q where At=∑i=1dM(i)min(t,d−i+1)A_t= _i=1^dM_(i) (t,d-i+1). Proof. By Lemma 3.1, we have ℛ~t≤∑s=2t[k^s∈Bts]. R_t≤Σ _s=2^tE_ x [ 1_\ k_s∈ B_t\ R_s ]. By Hölder’s inequality, we have [k^s∈Bts]≤ _ x [ 1_\ k_s∈ B_t\ R_s ]≤ [sp]1p[k^s∈Btq]1q _ x [ R^p_s ] 1pE_ x [ 1_ k_s∈ B_t^q ] 1q = = [sp]1p[k^s∈Bt]1q _ x [ R^p_s ] 1pE_ x [ 1_ k_s∈ B_t ] 1q for any p∈(1,12σ)p∈ (1, 12σ ) and q=p−1q= pp-1. Using Lemma F.1, we obtain [sp]1p[k^s∈Bt]1q≤[sp]1p[Bad(s,t)]1q _ x [ R^p_s ] 1pE_ x [ 1_ k_s∈ B_t ] 1q _ x [ R^p_s ] 1pE_ x [ 1_ Bad(s,t) ] 1q Now, using Lemma F.2, we have [sp]1p≤4(∑i=1d−sM(i)(p))1p _ x [ R^p_s ] 1p≤ 4 ( _i=1^d-sM^(p)_(i) ) 1p where M(j)(p):=[jp]M^(p)_(j):=E[ E_j^p], and M(1)(p)≥…≥M(d)(p)M^(p)_(1)≥...≥ M^(p)_(d) are the sorted p-moments. Hence, we have ℛ~t≤∑s=2t[sp]1p[Bad(s,t)]1q≤∑s=2t4(∑i=1d−sM(i)(p))1p[Bad(s,t)]1q. R_t≤Σ _s=2^tE_ x [ R^p_s ] 1pE_ x [ 1_ Bad(s,t) ] 1q≤Σ _s=2^t4 ( _i=1^d-sM^(p)_(i) ) 1pE_ x [ 1_ Bad(s,t) ] 1q. For any s≤ds≤ d, let Sd−s:=∑i=1d−sM(i)(p)S_d-s:= _i=1^d-sM^(p)_(i) and At=∑s=0t−1Sd−s=∑s=0t−1∑i=1d−sM(i)(p)=∑i=1dM(i)(p)min(t,d−i+1)A_t= _s=0^t-1S_d-s= _s=0^t-1 _i=1^d-sM^(p)_(i)= _i=1^dM^(p)_(i) (t,d-i+1). Then, ∑s=2t4(∑i=1d−sM(i)(p))1p[Bad(s,t)]1q≤At1/p(∑s=0t−1[Bad(s,t)])1/q=At1/p(∑s=0t−1ℙ(Bad(s,t)))1/qΣ _s=2^t4 ( _i=1^d-sM^(p)_(i) ) 1pE_ x [ 1_ Bad(s,t) ] 1q≤ A_t^1/p ( _s=0^t-1E_ x [ 1_ Bad(s,t) ] )^1/q=A_t^1/p ( _s=0^t-1P ( Bad(s,t) ) )^1/q by Hölder’s inequality. Now, combining this with Lemma F.3, we have the upper bound At1/p(∑s=0t−1ℙ(Bad(s,t)))1/q≤At1/p(mint,d−t,∑s=0t−1Z[ℙ(Epost(s,zm⋆))]+e−(t−s)ρPL(d−t))1/q A_t^1/p ( _s=0^t-1P ( Bad(s,t) ) )^1/q≤ A_t^1/p ( \t,d-t, _s=0^t-1E_Z [P (E_post(s,z_m ) ) ]+e^-(t-s) _PL(d-t) \ )^1/q which can in turn be bounded from above by At1/p(∑s=0t−1Z[ℙ(Epost(s,zm⋆))]+e−kρPL(d−t))1/q≤At1/p(mint,d−t,∑s=1tCλρλs+e−(t−s)ρPL(d−t))1/q A_t^1/p ( _s=0^t-1E_Z [P (E_post(s,z_m ) ) ]+e^-k _PL(d-t) )^1/q≤ A_t^1/p ( \t,d-t, _s=1^tC_λ _λ^s+e^-(t-s) _PL(d-t) \ )^1/q where Cλ:=(M−1)(1−εγεγpmin)λC_λ:=(M-1) ( 1- _γ _γp_ )^λ using Lemma F.4. The sum in the right-hand side term of this bound can be further upper bounded by Cλ1−ρλ+1eρ0−1. C_λ1- _λ+ 1e _0-1. This yields the final bound ℛ~t≤At1/p(mint,d−t,Cλ1−ρλ+1eρ0−1)1/q. R_t≤ A_t^1/p ( \t,d-t, C_λ1- _λ+ 1e _0-1 \ )^1/q. E.3 Proof of Theorem 3.5 Proof. From Lemma F.5, we have ℒ(^t,)−ℒ(t⋆,) ( P_t, x)-L( P _t, x) ≤ℒ~(^t,)−ℒ~(t⋆,) ≤ L( P_t, x)- L( P _t, x) +C′(|(,′)|peff+|(,′)|peff)∑j∈(,′)j∑j∈Γ(,′)j. +C ( | U( P, P )|p_eff+ | U( P, P )|p_eff ) _j∈ U( P, P ) E_j _j∈ ( P, P ) E_j. Taking expectations over x and using the Cauchy-Schwarz inequality, we obtain ℛt≤ℛ~t+C′((|(,′)|)peff+(|(,′)|)peff)∑j∈(,′)j∑j∈Γ(,′)j _t≤ R_t+C ( E_ x ( | U( P, P )| )p_eff+ E_ x (| U( P, P )| )p_eff )E_ x _j∈ U( P, P ) E_jE_ x _j∈ ( P, P ) E_j and using Jensen’s inequality, we obtain ℛt≤ℛ~t+C′(|(,′)|peff+(|(,′)|)peff)∑j∈(,′)j.∑j∈Γ(,′)j. _t≤ R_t+C ( E_ x| U( P, P )|p_eff+ E_ x (| U( P, P )| )p_eff ) E_ x _j∈ U( P, P ) E_j. E_ x _j∈ ( P, P ) E_j. We can in turn bound this using F.3 by ℛ~t+C′(|(,′)|peff+(|(,′)|)peff)∑j∈(,′)j R_t+C ( E_ x| U( P, P )|p_eff+ E_ x (| U( P, P )| )p_eff ) E_ x _j∈ U( P, P ) E_j ≤ ≤ ℛ~t+C′(d−t+Dtpeff+d−t+Dtpeff)Dt(d−t+Dt). R_t+C ( d-t+D_tp_eff+ d-t+D_tp_eff ) D_t(d-t+D_t). Appendix F Technical Lemmas F.1 Technical Lemma 1: Inclusion of Mismatch Events Recall that for any subset A⊆1,…,dA \1,…,d\, rkA(k) rk_A(k) is the rank of k E_k among all (j)j∈A( E_j)_j∈ A. Consider the event Bad(s,t):=rk^−s(k^s+1)>t−s=k^s+1<(t−s). Bad(s,t):= \ rk_ P_-s( k_s+1)>t-s \= \ E_ k_s+1< E_(t-s) \. Lemma F.1. We have ∑s=2tk^s∈Bts≤∑s=2tBad(s,t)s. Σ _s=2^t 1_\ k_s∈ B_t\ R_s≤Σ _s=2^t 1_ Bad(s,t) R_s. -10.0pt Proof. At time s≤ts≤ t, the test-time policy has selected at most s−1s-1 items from the top t−1t-1 values of (k)k=1d( E_k)_k=1^d, the most favorable case being when it selects the same items as the oracle policy. Hence, there are at least t−s−1t-s-1 items from the top t−1t-1 values of (k)k=1d( E_k)_k=1^d in the set ^−s P_-s. Therefore, if k^s∈Bt k_s∈ B_t, then by definition it is not in the top t−s−1t-s-1 items, because it would otherwise also belong to Gt=t⋆ _t= P _t B_t. Hence, k^s≤(t−s−1). E_ k_s≤ E_(t-s-1). We have hence proved that k^s∈Bt⊆Bad(s,t)\ k_s∈ B_t\ Bad(s,t), which proves our claim. F.2 Technical Lemma 2: Moment Bound on Conditional Deviations Lemma F.2. Let Mj(p):=[jp]M^(p)_j:=E[ E_j^p], and M(1)(p)≥…≥M(d)(p)M^(p)_(1)≥...≥ M^(p)_(d) be the sorted (p)(p)-moment order. Then, for every s and for any p∈(1,12σ)p∈ (1, 12σ ) [sp]1/p≤4(∑i=1d−sM(i))1pE[ R_s^p]^1/p≤ 4 ( _i=1^d-sM_(i) ) 1p Proof. Recall that s:=maxi,j∈^−t(i,s−j,s) R_s:= _i,j∈ P_-t( Y_i,s- Y_j,s). We have s≤maxi,j∈^−t(|i,s|+|j,s|)≤2maxi,j∈^−t|j,s|. R_s≤ _i,j∈ P_-t (| Y_i,s|+| Y_j,s| )≤ 2 _i,j∈ P_-t| Y_j,s|. Take p∈(1,12σ)p∈ (1, 12σ ). Since maxi,j∈^−t|j,s|≤(∑j∈^−t|j,s|p)1/p _i,j∈ P_-t| Y_j,s|≤ ( _j∈ P_-t| Y_j,s|^p )^1/p, we have sp≤2p∑j∈^−t|j,s|p R_s^p≤ 2^p _j∈ P_-t| Y_j,s|^p. Using the inequality for any k∈1,…,dk∈\1,…,d\ |k−[k∣ℱs]|p≤2p−1(|k|p+|[k∣ℱs]|p),| E_k-E[ E_k _s]|^p≤ 2^p-1 (| E_k|^p+|E[ E_k _s]|^p ), together with Jensen’s inequality, we obtain |j,s|p≤2p[|j|p]=2pMj(p)E| Y_j,s|^p≤ 2^pE[| E_j|^p]=2^pM^(p)_j. Therefore, [sp]≤2p∑j∈^−t|j,s|p≤4p∑j∈^−tMj(p).E[ R_s^p]≤ 2^p _j∈ P_-tE| Y_j,s|^p≤ 4^p _j∈ P_-tM^(p)_j. Since |^−t|=d−s| P_-t|=d-s deterministically and ∑j∈^−tMj(p)≤∑i=1d−sM(i)(p) _j∈ P_-tM^(p)_j≤ _i=1^d-sM^(p)_(i), the claimed bound follows. F.3 Technical Lemma 3: Bounding Disagreement Probability For each state m∈1,…,Mm∈\1,…,M\, define uj(m):=logaj+2ηj+2βjzm,θj(m):=exp(uj(m)2σ). u_j^(m):= a_j+2 _j+2 _jz_m, _j^(m):= ( u_j^(m)2σ ). Also let γ:=minm∈[M]minj≠j′|uj(m)−uj′(m)|>0γ:= _m∈[M] _j≠ j |u_j^(m)-u_j ^(m) |>0 and Rmax:=maxj∈[d]maxme2βjzmminme2βjzm<∞R_ := _j∈[d] _me^2 _jz_m _me^2 _jz_m<∞. Finally, define εγ:=eγ−1Rmax+eγ−1∈(0,1) _γ:= e^γ-1R_ +e^γ-1∈(0,1) and ρPL(k):=minm∈[M]minJ⊂[d]|J|=kmaxj∈Jθj(m)∑ℓ∈Jθℓ(m)≥1k _PL(k):= _m∈[M] _ subarraycJ⊂[d]\\ |J|=k subarray _j∈ J _j^(m) _ ∈ J _ ^(m)≥ 1k. Lastly, define αm(ℱs):=ℙ[Z=zm∣ℱs] _m(F_s):=P [Z=z_m _s ] and Epost(s,zm):= E_post(s,z_m):= ℙ[Z=zm∣ℱs]<1−εγ∣Z=zm, \P [Z=z_m _s ]<1- _γ Z=z_m \, Epost(s,Z):= E_post(s,Z):= ℙ[Z=zm∣ℱs]<1−εγ∣Z. \P [Z=z_m _s ]<1- _γ Z \. Lemma F.3. Assume that ℙ[Z=zm∣ℱs]>0P [Z=z_m _s ]>0 for all zmz_m and s≥1s≥ 1. We have ℙ(Bad(s,t)) ( Bad(s,t)) ≤Z[ℙ(Epost(s,Z))]+e−kρPL(d−t). _Z [P (E_post(s,Z) ) ]+e^-k _PL(d-t). Proof. Let us first work conditionally on Z=zm⋆Z=z_m . Define j⋆(zm⋆):=argmaxj∈^−tuj(m⋆)j (z_m ):= _j∈ P_-tu_j^(m ). Since γ>0γ>0, this index is uniquely defined. We claim that on Epost(s,zm⋆)cE_post(s,z_m )^c, the test-time optimal policy selects k^s+1=j⋆(zm⋆) k_s+1=j (z_m ). To see this, note that [j∣ℱs]∝elogaj+2ηj∑m=1Mαm(ℱs)e2βjzmE[ E_j _s] e a_j+2 _j _m=1^M _m(F_s)e^2 _jz_m. Define Tj:=∑m=1Mαm(ℱs)e2βjzmT_j:= _m=1^M _m(F_s)e^2 _jz_m. On Epost(s,zm⋆)cE_post(s,z_m )^c, we have αm⋆(ℱs)≥1−εγ _m (F_s)≥ 1- _γ. Hence: Tj=αm⋆(ℱs)e2βjzm⋆+∑m≠m⋆αm(ℱs)e2βjzm⏟>0>(1−εγ)e2βjzm⋆ T_j= _m (F_s)e^2 _jz_m + _m≠ m _m(F_s)e^2 _jz_m_>0>(1- _γ)e^2 _jz_m where the strict inequality uses our assumption ℙ[Z=zm∣ℱs]>0P [Z=z_m _s ]>0. On the other hand, Tj T_j ≤(1−εγ)e2βjzm⋆+εγRmaxe2βjzm⋆=e2βjzm⋆(1−εγ+εγRmax). ≤(1- _γ)e^2 _jz_m + _γR_ e^2 _jz_m =e^2 _jz_m (1- _γ+ _γR_ ). Thus, up to a positive multiplicative constant: (1−εγ)euj(m⋆)≤[j∣ℱs]≤(1−εγ+εγRmax)euj(m⋆)(1- _γ)e^u_j^(m ) [ E_j _s]≤ (1- _γ+ _γR_ )e^u_j^(m ). For any j≠j⋆(zm⋆)j≠ j (z_m ) in ^−t P_-t: [j⋆∣ℱs][j∣ℱs] E[ E_j _s]E[ E_j _s] >euj⋆(m⋆)−uj(m⋆)1−εγ1−εγ+εγRmax≥eγ1−εγ1−εγ+εγRmax, >e^u_j ^(m )-u_j^(m ) 1- _γ1- _γ+ _γR_ ≥ e^γ 1- _γ1- _γ+ _γR_ , and this ratio exceeds 1 by definition of εγ _γ, proving our claim. From Assumption 3.2, conditional on Z=zm⋆Z=z_m , we have logj=uj(m⋆)+ξj E_j=u_j^(m )+ _j where ξj∼Gumbel(0,2σ) _j (0,2σ). The ranking follows a Plackett–Luce distribution with weights θj(m⋆)\ _j^(m )\ (Yellott Jr, 1977; Ma et al., 2021). Under the PL sequential sampling view, the probability that j⋆(zm⋆)j (z_m ) is not selected in the first s draws is bounded as ℙ(rk^−s(j⋆)>t−s∣Z=zm⋆)≤e−sp⋆(zm⋆)≤e−sρPL(d−s)P ( rk_ P_-s(j )>t-s Z=z_m )≤ e^-sp (z_m )≤ e^-s _PL(d-s), where p⋆(zm⋆):=θj⋆(m⋆)/∑ℓ∈Jsθℓ(m⋆)≥ρPL(d−s)p (z_m ):= _j ^(m )/ _ ∈ J_s _ ^(m )≥ _PL(d-s). This bound does not depend on zm⋆z_m , so we may average over Z. Finally, decomposing Bad(s,t) Bad(s,t) by latent states and conditioning on Epost(s,zm⋆)E_post(s,z_m ) vs. its complement: ℙ(Bad(s,t))≤ℙ(Epost(s))+e−kρPL(d−t). ( Bad(s,t)) (E_post(s))+e^-k _PL(d-t). F.4 Technical Lemma 4: Posterior Concentration Recall that given Z=zmZ=z_m, we have Xj∼Gumbel(ηj+βjzm,σ)X_j ( _j+ _jz_m,σ) with density fj,mf_j,m. Fix λ∈(0,1)λ∈(0,1) and define the Chernoff coefficient (Nielsen, 2011) between states m and m′m for feature j as cj(m,m′,λ):=∫fj,m(x)1−λfj,m′(x)λxc_j(m,m ;λ):= f_j,m(x)^1-λf_j,m (x)^λ\,dx. For two Gumbel distributions with locations μ,μ′μ,μ and common scale σ, this has the closed form c(μ,μ′,λ)=e−λΔ(1−λ)+λe−Δc(μ,μ ;λ)= e^-λ (1-λ)+λ e^- , where Δ:=μ−μ′σ := μ-μ σ. Let ρλ:=maxj∈[d]maxm≠m′cj(m,m′,λ)<1 _λ:= _j∈[d] _m≠ m c_j(m,m ;λ)<1 and pmin:=minm∈[M]pm>0p_ := _m∈[M]p_m>0. Lemma F.4. For any value of Z=zm⋆Z=z_m , any ε∈(0,1) ∈(0,1), any s≥1s≥ 1, and λ≥1λ≥ 1, ℙ(αm⋆(ℱs)≤1−ε∣Z=zm⋆)≤(M−1)(1−εpmin)λρλs.P ( _m (F_s)≤ 1- Z=z_m )≤(M-1) ( 1- p_ )^λ _λ^s. In particular, setting ε=εγ = _γ, we have ℙ(Epost(s,zm⋆))≤CλρλsP (E_post(s,z_m ) )≤ C_λ _λ^s, where Cλ:=(M−1)(1−εγεγpmin)λC_λ:=(M-1) ( 1- _γ _γp_ )^λ. Proof. For any state m≠m⋆m≠ m , define the likelihood ratio κs(m):=pmpm⋆∏r=1sfir,m(Xir)fir,m⋆(Xir) _s(m):= p_mp_m _r=1^s f_i_r,m(X_i_r)f_i_r,m (X_i_r), where Xir=logxirX_i_r= x_i_r. By Bayes’ rule, αm⋆(ℱs)=11+∑m≠m⋆κs(m) _m (F_s)= 11+ _m≠ m _s(m). If αm⋆(ℱs)≤1−ε _m (F_s)≤ 1- , then ∑m≠m⋆κs(m)≥ε1−ε _m≠ m _s(m)≥ 1- . By the pigeonhole principle, there exists some m≠m⋆m≠ m such that κs(m)≥ε(1−ε)(M−1) _s(m)≥ (1- )(M-1). By a union bound and Markov’s inequality with exponent λ≥1λ≥ 1: ℙ(αm⋆(ℱs)≤1−ε∣Z=zm⋆)≤∑m≠m⋆[κs(m)λ∣Z=zm⋆](ε(1−ε)(M−1))λ. ( _m (F_s)≤ 1- Z=z_m )≤ _m≠ m E[ _s(m)^λ Z=z_m ] ( (1- )(M-1) )^λ. Since Xi1,…,XisX_i_1,…,X_i_s are independent draws from fi1,m⋆,…,fis,m⋆f_i_1,m ,…,f_i_s,m given Z=zm⋆Z=z_m , each factor satisfies [(fir,m(Xir)fir,m⋆(Xir))λ|Z=zm⋆]=cir(m⋆,m,λ)≤ρλE [ ( f_i_r,m(X_i_r)f_i_r,m (X_i_r) )^λ\, |\,Z=z_m ]=c_i_r(m ,m;λ)≤ _λ. Therefore, [κs(m)λ∣Z=zm⋆]≤pmin−λρλsE [ _s(m)^λ Z=z_m ]≤ p_ ^-λ _λ^s. Summing over all M−1M-1 wrong states yields the claim. Since the bound does not depend on zm⋆z_m , it holds for any latent state. F.5 Technical Lemma 5: Spectral Misalignment Bound Define for any subset ⊂1,…,d P⊂\1,…,d\: ℳ():=ℒ()−ℒ~()M( P):=L( P)- L( P) and u():=[j∉∥j∥j]j=1du( P):= [1_\j∉ P\\|w_j\| x_j ]_j=1^d. Lemma F.5. For any two sets ,′ P, P , let (,′):=(∩′)c U( P, P ):= ( P∩ P )^c and Γ(,′)=Δ′ ( P, P )= P P . We have |ℳ()−ℳ(′)|≤C′(|(,′)|peff+|(,′)|peff)∑j∈(,′)j∑j∈Γ(,′)j|M( P)-M( P )|≤ C ( | U( P, P )|p_eff+ | U( P, P )|p_eff ) _j∈ U( P, P ) E_j _j∈ ( P, P ) E_j Proof. Let B:=[j‖j‖]j=1dB:= [ w_j\|w_j\| ]_j=1^d and define H:=BTB−IH:=B^TB-I. We have ℳ()−ℳ(′)=(u()−u(′))⊤HU(u()+u(′))M( P)-M( P )=(u( P)-u( P )) H_U(u( P)+u( P )) where HU:=BU⊤BU−IH_U:=B_U B_U-I is the restriction of H to (,′) U( P, P ). Therefore: |ℳ()−ℳ(′)| |M( P)-M( P )| ≤‖HU‖op‖u()−u(′)‖2‖u()+u(′)‖2 ≤\|H_U\|_op\|u( P)-u( P )\|_2\|u( P)+u( P )\|_2 ≤‖HU‖op‖u()−u(′)‖2(‖u()‖2+‖u(′)‖2). ≤\|H_U\|_op\|u( P)-u( P )\|_2(\|u( P)\|_2+\|u( P )\|_2). Note that |(,′)|=d−t+kt| U( P, P )|=d-t+k_t, and ‖u()−u(′)‖22=∑j∈Γ(,′)‖wj‖22xj2=∑j∈(,′)Vj\|u( P)-u( P )\|_2^2= _j∈ ( P, P )\|w_j\|_2^2x_j^2= _j∈ U( P, P )V_j. Using ‖u()+u(′)‖22≤4∑j∈U(,′)j\|u( P)+u( P )\|_2^2≤ 4 _j∈ U( P, P ) E_j and Assumption 3.4 concludes the proof. Appendix G Experimental Details G.1 Datasets We evaluate echo-k echo -k and all baselines on six datasets spanning seven downstream tasks. Table 5 provides a summary. Below we describe each dataset in detail, including the downstream task, preprocessing choices, and the specific aspects that make it relevant for evaluating budgeted feature acquisition. Table 5: Overview of datasets used for experiments. Dataset Task Train/Val/Test # classes # views Source Synthetic Random isotropic regression 200/-/200 — 64 — NSCLC patient cohort Cancer grading 1.6k/400/400 3 40 Cords et al. (2024) Cancer subtyping 2 40 Cords et al. (2024) hPancreas Cell type annotation 9.5k/1k/4.2k 14 200 Ding et al. (2025) MiniBooNE Particle identification 65k/5k/6k 2 33 Roe (2005) MNIST Image classification 50k/10k/10k 10 196 LeCun et al. (1998) Fashion-MNIST Image classification 50k/10k/10k 10 196 Xiao et al. (2017) MMEarth-Bench Soil pH regression 5.7k/1.2k/350 — 12 Gordon et al. (2026) G.1.1 Vision Datasets: MNIST and Fashion-MNIST MNIST (LeCun et al., 1998) and Fashion-MNIST (Xiao et al., 2017) are standard computer vision benchmarks consisting of images of handwritten digits and Zalando article images, respectively. Both have 10-class classification as the downstream task. Following Li and Oliva (2021), we downsample images from 28×2828× 28 to 14×1414× 14 resolution, treating each pixel as an independently acquirable feature and yielding d=196d=196 acquisition units. This downsampling increases the information density per pixel and makes the acquisition problem more challenging, since fewer pixels suffice to reach a given accuracy. We use a 70:15:15 split (50k/10k/10k). To ensure a controlled comparison, all acquisition methods are initialized from the same top-corner pixel ( ), so differences in accuracy reflect only the quality of the subsequent acquisition strategy. G.1.2 Physics: MiniBooNE The MiniBooNE dataset (Roe, 2005) originates from the Fermilab MiniBooNE neutrino experiment. The goal is binary classification: distinguishing electron-neutrino signal events from muon-neutrino background events. The dataset contains d=33d=33 reconstructed physics variables derived from detector signals, including Cherenkov ring characteristics, event energy, and track reconstruction outputs. In practice, computing these reconstructed variables from raw detector data is expensive and the variables exhibit substantial mutual correlations, making it undesirable to rely on the full feature set at test time. We adopt the preprocessing protocol of Norcliffe et al. (2025), which standardizes features and removes duplicate events, and use a 70:15:15 split (65k/5k/6k). This dataset is particularly relevant for evaluating echo-k echo -k because the strong inter-feature correlations create a setting where greedy, myopic acquisition can be suboptimal: selecting the individually most informative variable first may be redundant with the second-most informative one, motivating the long-horizon planning that echo-k echo -k provides. G.1.3 Single-Cell Transcriptomics: hPancreas The hPancreas dataset (Ding et al., 2025) is a curated single-cell RNA-sequencing collection from human pancreatic tissue with 14 cell type labels spanning endocrine, exocrine, and immune populations. This dataset is widely used as a benchmark in the single-cell foundation model literature. From the 1,200 genes available, we retain the top 200 most variable genes using Cell Ranger (Zheng et al., 2017) and ScanPy (Wolf et al., 2018), following standard practice for reducing noise from uninformative transcripts while preserving biologically meaningful variation. We use the preprocessing, gene filtering, and data splits provided by (Ding et al., 2025) (9.5k/1k/4.2k). Cell-type annotation serves as a representative downstream task: it is the most common evaluation objective in single-cell analysis and requires the acquired gene subset to preserve fine-grained phenotypic distinctions between closely related cell populations. Crucially, the echo-k echo -k policy is trained without any access to cell-type labels; the annotation task is used purely for post-hoc evaluation of the acquired gene panels. G.1.4 Spatial Proteomics: NSCLC Patient Cohort The NSCLC clinical cohort (Cords et al., 2024) consists of spatial proteomics data acquired via imaging mass cytometry (IMC) at 1μm1\, /px resolution with d=40d=40 semantically meaningful protein markers. The marker panel includes tumor markers (e.g., panCK), proliferation markers (e.g., Ki-67), immune cell markers (e.g., CD4, CD8, CD68), stromal markers, and vascular markers, providing a comprehensive view of the tumor microenvironment. Clinical annotations include cancer subtyping (2 classes: adenocarcinoma vs. squamous cell carcinoma) and cancer grading (3 classes), the latter being substantially more challenging due to subtler morphological and molecular differences between grades. We adopt the preprocessing pipeline described in (Wenckstern et al., 2025), which includes cell segmentation, marker normalization, and quality filtering. The experimental setup is inherently sequential: echo-k echo -k iteratively selects one protein marker at a time, the VirTues backbone updates its tissue embedding given the newly observed marker, and the updated embedding informs the next acquisition decision. This mirrors the real-world workflow in multiplexed imaging, where tissue sections are characterized through successive rounds of antibody staining and each additional round carries a cost. The acquisition budget is set to T=10T=10 markers (25% of the full panel), reflecting realistic constraints on antibody panel size in clinical multiplex imaging. G.1.5 Geospatial Earth Observations: MMEarth-Bench The MMEarth-Bench dataset (Gordon et al., 2026) is a global multimodal Earth-observation benchmark to evaluate models under heterogeneous sensing conditions. Each location is represented by 12 aligned geospatial views, including optical satellite imagery, radar measurements, terrain and elevation products, land-cover maps, canopy-height information, meteorological variables, and geolocation metadata. It provides a multimodal acquisition setting: each view corresponds to a distinct sensing product or environmental data source that may differ in availability, cost, spatial resolution, and information content. We treat each modality group as one acquisition unit and pair echo-k echo -k with a frozen multimodal geospatial representation model trained on the full set of available views. We evaluate downstream transfer on soil pH regression, reporting R2R^2 as a function of the acquired modality budget. G.1.6 Synthetic Dataset Figure 5: Overview of the Latent Gumbel Correlation Model. a. Distribution of log(j) ( x_j) for 3 equiprobable latent states Z with support −1,0,1\-1,0,1\. b. Correlation of the features under the gradient correlation family described in § G.1.6. We generate synthetic data under the data generating process (DGP) of Assumption 3.2. We consider a gradient correlation family shared by a common latent state Z, where Z∼Discrete(zmm=1M,pmm=1M)Z (\z_m\_m=1^M,\p_m\_m=1^M ), εj∼Gumbel(0,σ) _j (0,σ) being independent noise. Each feature is generated as Xj∼ηj+βjZ+εjX_j _j+ _jZ+ _j, where ηj _j and βj _j are deterministic coefficients, and we develop a gradient-like correlation structure by letting βj:=βmax(1−αj−1d−1), _j:= _ (1-α j-1d-1 ), and set xj=eXjx_j=e^X_j. In Fig. 5a, we show the data distribution when the support of Z has cardinality 3, and zm∈−1,0,1z_m∈\-1,0,1\. Here βmax _ is the maximum strength and α∈[0,1]α∈[0,1] controls the decay rate. Under this construction, early features exhibit strong mutual correlations through Z, while later features are increasingly dominated by independent noise. The induced correlation structure is: Corr(Xi,Xj)=βiβjVar(Z)(βj2Var(Z)+π2σ26)(βk2Var(Z)+π2σ26).Corr(X_i,X_j)= _i _jVar(Z) ( _j^2Var(Z)+ π^2σ^26 ) ( _k^2Var(Z)+ π^2σ^26 ). Regret bound experiment (Fig. 2 We set p=25,d=32p=25,d=32 (i.e., ∈ℝp×d W ^p× d), σ=0.2σ=0.2, M=5M=5 with zm∈−1,−1/2,0,1/2,1z_m∈\-1,-1/2,0,1/2,1\ and pm=0.2∀mp_m=0.2\;∀ m. We consider ηj=j−1d−1−12 _j= j-1d-1- 12 for j∈1,…,dj∈ 1,…,d. For the gradient correlation family, we set βmax=1.5 _ =1.5 and α=0.9α=0.9. Random isotropic regression experiment (Fig. 6). We increase the acquisition horizon by setting d=64d=64 and p=50p=50, keeping all other hyperparameters fixed. We sample the ground-truth regression weights as ∗∼(0,1dI) W^* (0, 1dI) and generate labels as =∗+′ y= W^* x+ where ′∈ℝp ^p has i.i.d. entries sampled from Gumbel(0,1)Gumbel(0,1). G.2 Architectures and Backbone Details G.2.1 Masked Autoencoder (MNIST, Fashion-MNIST, MiniBooNE) For image and tabular data, we train a masked autoencoder via self-supervised learning on the full dataset. Given an input x, a random subset of features is revealed via a binary mask m∈0,1dm∈\0,1\^d, and the model learns an encoder z=fθ(x⊙m,m)z=f_θ(x m,m) and decoder x^=gθ(z) x=g_θ(z) to reconstruct the full input using an MSE objective. The encoder receives both the masked input and the mask itself, allowing it to distinguish between zero-valued features and unobserved ones. The masking ratio is sampled uniformly during training to expose the model to a wide range of partial observations. After pretraining, the encoder is frozen and used as the representation backbone for echo-k echo -k. For downstream evaluation, a linear probe is trained on z for classification under random masking to simulate partial observations; this ensures that the probe is calibrated to the type of partial inputs encountered during sequential acquisition. The same frozen encoder is shared across all baselines and echo-k echo -k. G.2.2 TABULA Foundation Model For single-cell RNA-seq, we use the pretrained TABULA foundation model (Ding et al., 2025), a self-supervised transformer designed for single-cell transcriptomic data. TABULA is trained on large-scale single-cell atlases and can embed partially observed gene expression profiles into cell-level representations. When a subset of genes is unobserved, TABULA treats these entries as missing and produces an embedding conditioned only on the available genes. This native support for partial observations makes it well-suited as a backbone for echo-k echo -k. The downstream task of cell-type annotation is evaluated via a linear probe trained on top of the cell embeddings. We use TABULA without any fine-tuning; the model weights are frozen throughout both policy learning and evaluation. G.2.3 VirTues Foundation Model For the NSCLC patient cohort, we deploy echo-k echo -k with the VirTues spatial proteomics foundation model (Wenckstern et al., 2025). VirTues is a graph-based transformer that operates on cell-level protein expression profiles within tissue sections, producing both cell-level and tissue-level representations. Tissue-level representations are obtained by averaging the cell-level summary tokens across all cells in the tissue sample. When a protein marker is unobserved, its expression value is treated as missing in the cell-level input, and VirTues produces an embedding conditioned on the available markers. Linear probes are trained for cancer subtyping and grading using the tissue-level embeddings. Importantly, the same echo-k echo -k policy is used for both downstream tasks: the acquisition order is determined once by the policy and then evaluated on both tasks, demonstrating task-agnostic panel design. G.2.4 GeoSpatial Encoder For MMEarth-Bench, a masked encoder is used. Concretely, each of the 12 Earth-observation modalities is first converted to a modality-level vector by global average pooling for spatial inputs and identity for tile-level inputs, then projected to a shared dimension and multiplied by a binary modality mask indicating which modalities are observed. The masked per-modality projections are concatenated together with the mask itself and passed through an MLP encoder to produce a latent representation z. This encoder is trained under random modality masking, so it learns to make reconstructions from arbitrary partial modality subsets while explicitly distinguishing unobserved modalities from zero-valued features. Backbone-agnosticism. Across all settings, both echo-k echo -k and the baselines have access only to the pretrained models during the feature selection phase. Linear probes are used exclusively for downstream evaluation after feature acquisition. echo-k echo -k interfaces with each backbone only through the partial embedding (St) z(S_t) and the full-information target ⋆ z ; no architecture-specific modifications are needed. This design means that echo-k echo -k can be deployed with any new backbone simply by providing the embedding function, without rewriting any acquisition logic. G.3 Evaluation Protocol Each dataset is partitioned into training, validation, and test splits. For image and tabular tasks, we use a 70:15:15 split. For cell-type annotation, we adopt the predefined splits released with TABULA (Ding et al., 2025), and for cancer subtyping and grading, we follow the test splits defined in (Wenckstern et al., 2025). Acquisition curves. The primary evaluation metric is the acquisition curve: for each method, we plot downstream performance (MSE or classification accuracy) as a function of the number of features acquired t∈[[1,d]]t∈[\![1,d]\!]. This allows us to compare methods across the full range of acquisition budgets and identify regimes where differences are most pronounced. In practice, the early regime (5–25% of features) is of greatest interest, since this is where cost savings are largest. Linear probes. For all real-world datasets, downstream performance is evaluated by training a linear probe on the partial embeddings (St) z(S_t) produced by the frozen backbone. A single linear probe is trained per downstream task using random masking augmentation and held fixed across acquisition budgets. This ensures that the evaluation is task-specific (it depends on the downstream labels) while the acquisition policy remains task-agnostic. Reproducibility. All experiments are run with three random seeds for both the echo-k echo -k policy and the downstream evaluation. The Random baseline is additionally averaged over five independent runs per seed to reduce variance. Error bands in all figures correspond to standard error across seeds. G.4 Baselines We compare echo-k echo -k against a total of 15 baselines spanning synthetic (S), image (I), tabular (T), single-cell RNA-seq (C), and multiplex imaging (M) datasets. Below, we describe each baseline and indicate the settings in which it is evaluated. G.4.1 Oracle and Random Baselines Oracle (S, I, T). A one-step myopic oracle that greedily selects the next feature maximizing the immediate reduction in distance between the current embedding and the full (gold) embedding. This policy is anti-causal: it has access to unobserved features and corresponds to the oracle analyzed in our theoretical results (§3). It serves as an upper bound on greedy acquisition performance but is not deployable at test time. Random (S, I, T, C, M). Selects the next view uniformly at random from the set of unobserved features. Provides a minimal reference point for all experiments. G.4.2 Model-Driven Heuristic Baselines Prior (S). Specific to the synthetic setting. Ranks features according to their unconditional expected contribution [Ej]E[E_j], which under Assumption 3.2 admits the closed form: [j]=‖j‖22Γ(1−2σ)e2ηj∑m=1Mpme2βjzm∝‖j‖22e2ηj∑m=1Mpme2βjzm.E[ E_j]=\|w_j\|_2^2 (1-2σ)e^2 _j _m=1^Mp_me^2 _jz_m \|w_j\|_2^2e^2 _j _m=1^Mp_me^2 _jz_m. Unlike echo-k echo -k, this baseline does not condition on observed features and therefore captures only population-level feature importance. GIR (I, T, S, M). Global Impact Ranking is a practical extension of Prior that does not rely on a known generative model. For each feature i, a validation-set score si:=‖full−i‖2s_i:=\| z_full- z_-i\|_2 is computed, where full z_full denotes the self-supervised embedding obtained using all features and −i z_-i denotes the embedding obtained when feature i is withheld. This induces a fixed, instance-independent acquisition order that captures population-level importance but not individual-level adaptivity. ESM-Max, ESM-Min (M). For multiplex imaging only. Protein markers are embedded using ESM2 (Lin et al., 2023). For each marker, pairwise Euclidean distances are computed in the ESM2 embedding space. Markers are ranked by either minimum (ESM-Min) or maximum (ESM-Max) distance, yielding fixed acquisition orders based on semantic similarity or dissimilarity, following (Jain et al., 2025). G.4.3 Unsupervised Feature Selection Baselines The following methods were originally designed to identify informative feature subsets given full feature access (Solorio-Fernández et al., 2020). We adapt them to the sequential acquisition setting by using their induced feature rankings as acquisition orders. While not designed for instance-adaptive or sequential settings, they provide strong unsupervised baselines for ranking-based selection. All scores are computed on the validation set and induce a fixed acquisition order applied at test time. Variance (I, T, S). Ranks features by their marginal variance over the validation set, selecting in decreasing order of variance. spec-φ3 _3 (I, T). Ranks features based on alignment with the leading k−1k\!-\!1 eigenvectors of the graph Laplacian, where k is the number of clusters (Zhao and Liu, 2007). Features with stronger alignment to these optimal soft cluster indicators receive higher scores. Laplacian Score (I, T). The Laplacian Score (He et al., 2005) evaluates features by how well they preserve local geometric structure encoded by a nearest-neighbor graph. For each feature r, it computes Lr=f~r⊤Lf~rf~r⊤Df~rL_r= f_r L f_r f_r D f_r, where L=D−SL=D-S is the graph Laplacian, S is the similarity matrix, and f~r f_r is the mean-centered weighted feature vector. Features with smaller scores are ranked higher. MCFS (I, T). Multi-Cluster Feature Selection (Cai et al., 2010) selects features via spectral analysis of the graph Laplacian. It first solves the generalized eigenvalue problem Ly=λDyLy=λ Dy to obtain K eigenvectors Y=[y1,…,yK]Y=[y_1,…,y_K]. For each yky_k, it solves an ℓ1 _1-regularized regression minak‖yk−XTak‖2+β|ak| _a_k\|y_k-X^Ta_k\|^2+β|a_k| to obtain sparse coefficients. Features are ranked by MCFS(j)=maxk|ak,j| MCFS(j)= _k|a_k,j| and selected in decreasing order. UDFS (I, T). Unsupervised Discriminative Feature Selection (Yang et al., 2011) learns a sparse projection matrix W∈ℝd×cW ^d× c by minimizing local discriminative scores with ℓ2,1 _2,1-norm regularization: minWTW=ITr(WTMW)+γ‖W‖2,1 _W^TW=ITr(W^TMW)+γ\|W\|_2,1, where M encodes local neighborhood structure via k-nearest neighbors. Features are ranked by the row norms ‖wi‖2\|w_i\|_2 of W in descending order. SVD-Ent (I, T). SVD-Entropy (Varshavsky et al., 2006) ranks features by their contribution to SVD-entropy. For each feature i, its contribution is CEi=E(A)−E(A−i)CE_i=E(A)-E(A_-i), where E=−1logN∑j=1NVjlogVjE=- 1 N _j=1^NV_j V_j and Vj=sj2/∑ksk2V_j=s_j^2/ _ks_k^2 are normalized squared singular values. Features with larger CEiCE_i are ranked higher. G.4.4 Adaptive Label-Free Baselines The baselines above induce fixed, instance-independent acquisition orders. To compare against policies that condition acquisition on the current instance, we additionally evaluate the following label-free adaptive baselines on MiniBooNE and hPancreas (§ H.3). As for echo-k echo -k, these methods observe only the currently acquired values and the acquisition mask at deployment, and never access the full-view embedding or downstream labels. GSMRL-AIR (T, C). An instance-adaptive RL method based on acquisition with generative surrogate models (Li and Oliva, 2021), adapted to the label-free setting. The policy conditions its action selection on the per-sample conditional imputations of unobserved features produced by a generative surrogate, so that acquisition decisions depend on the values observed for each instance. ALG (T, C). Adaptive Latent Gain is a learned regressor that predicts, for each unobserved view, the conditional one-step improvement in representation recovery directly from the current partial observation, and greedily acquires the view with the largest predicted gain. ALG shares the reward and state representation of echo-k echo -k, but optimizes only the immediate gain rather than a long-horizon objective, and therefore isolates the contribution of sequential planning. CIG (T). Cluster Information Gain fits a Gaussian mixture model to the training data and scores each unobserved feature by the mutual information between the feature and the mixture component under the posterior given the currently observed features. The feature with the largest information gain is acquired next. Evaluated on MiniBooNE only. G.4.5 Summary Table Table 6: Summary description of implemented baselines. Methods marked with (†)( ) rely on unsupervised feature selection and have oracle access to all features at ranking time. Methods marked with a (⋆)( ) are only deployed on the cancer grading and typing datasets, since they rely on a protein sequence foundation model (Lin et al., 2023). Methods marked with (‡)( ) are label-free adaptive baselines that select instance-specific feature sets through continuous updates of their selection policies. Name Model Driven? Selection Criterion Adaptive? Source Oracle ✓ 1-step Oracle knowledge ✓ —- GIR ✓ Average contribution to latent state ✗ —- Random ✗ Uniform distribution over unseen features ✗ —- Laplacian Score † ✗ Locality preservation ✗ He et al. (2005) spec-φ3 _3 † ✗ Graph-based ranking ✗ Zhao and Liu (2007) UDFS † ✗ Local discriminative structure ✗ Yang et al. (2011) MCFS † ✗ Multi-cluster structure ✗ Cai et al. (2010) SVD-Ent † ✗ SVD-entropy contribution ✗ Varshavsky et al. (2006) Variance † ✗ Maximum variance ✗ —- ESM-Max (⋆ ) ✓ ESM2-based ✗ Lin et al. (2023) ESM-Min (⋆ ) ✓ ESM2-based ✗ Lin et al. (2023) GSMRL-AIR (‡ ) ✗ Generative surrogate imputations ✓ Li and Oliva (2021) ALG (‡ ) ✓ Predicted one-step latent improvement ✓ —- CIG (‡ ) ✗ GMM mutual information ✓ —- echo-k echo -k ✓ Learned policy via expected latent improvement ✓ This paper G.5 Implementation Details of echo-k echo -k We cast sequential feature acquisition in echo-k echo -k as a finite-horizon decision problem over partial observations, closely aligned with the conditional surrogate objective analyzed in § 3. The goal of the agent is to sequentially select features so as to minimize the discrepancy between the latent representation induced by the acquired subset and the full-information latent representation of a frozen self-supervised model. G.5.1 MDP Formulation Let d denote the total number of features and T the acquisition budget. An episode corresponds to sequentially selecting T features without replacement. At step t, the agent has observed a subset St⊆1,…,dS_t \1,…,d\. We define a finite-horizon Markov decision process (,,P,r)(S,A,P,r) as follows. State. The observable state summarizes the acquisition history and the induced latent representation: st:=(St,(St),t/T,‖(St)‖2,⊙t)s_t:= (S_t, z(S_t),t/T,\| z(S_t)\|_2, x m_t ) where t∈0,1d m_t∈\0,1\^d is the binary acquisition mask corresponding to StS_t, and (St) z(S_t) denotes the latent embedding produced by the frozen self-supervised encoder given the partially observed input. This state representation renders the process Markov with respect to the surrogate objective. Action. At each step, the agent selects a feature index at∈t:=1,…,d∖Sta_t _t:=\1,…,d\ S_t, corresponding to acquiring a previously unobserved feature. We apply invalid action masking (Huang and Ontañón, 2020), assigning zero probability to actions in StS_t during policy evaluation and optimization. Transition. Transitions are deterministic: selecting at=ja_t=j yields St+1=St∪jS_t+1=S_t∪\j\ and updates the latent embedding to (St+1) z(S_t+1). G.5.2 Reward Design Let ∗ z^* denote the latent embedding obtained from the full feature set. Consistent with the surrogate loss analyzed in § 3, we define the instantaneous reward as the normalized reduction in representation error: rt=(1+1t<t0λ)(‖∗−(S)‖2−‖∗−(S∪j)‖2‖∗−(S0)‖2),r_t=(1+1\t<t_0\λ) ( \| z^*- z(S)\|_2-\| z^*- z(S∪\j\)\|_2\| z^*- z(S_0)\|_2 ), where λ>0λ>0 is the gain factor and S0S_0 denotes the singleton initial feature set. This reward is a shaped version of the marginal gain Δ(at∣St)=‖∗−(St)‖2−‖∗−(St∪at)‖2 (a_t S_t)=\| z^*- z(S_t)\|_2-\| z^*- z(S_t∪\a_t\)\|_2, which serves as a proxy for the conditional expected contribution [Eat∣ℱt]E[E_a_t _t] appearing in our theoretical analysis. The step-dependent amplification factor (1+λ1t<t0)(1+ 1\t<t_0\) encourages refinement in early acquisition stages, where marginal reductions are typically largest. G.5.3 Policy Learning We learn a stochastic policy πθ(at∣st) _θ(a_t s_t) using PPO (Schulman et al., 2017) with invalid action masking (Huang and Ontañón, 2020). During both training and inference, the policy has access only to the frozen self-supervised encoder and the induced latent representations. Downstream task labels and linear probes are used exclusively for post-hoc evaluation and do not influence the policy or reward. G.5.4 Hyperparameters We implement echo-k echo -k using the Stable-Baselines3 (SB3) (Raffin et al., 2021) framework with sb3-contrib for support of masked action spaces. Unless mentioned, all PPO parameters are default of SB3. All experiments were conducted on NVIDIA A100 GPUs (80GB) with 128 AMD EPYC 7543 32-core processors. Table 7 summarizes the key hyperparameters across datasets. Table 7: Hyperparameters used for echo-k echo -k across datasets. The remaining hyperparameters follow the defaults in Stable-Baselines3. Parameter MNIST/Fashion-MNIST MiniBooNE hPancreas NSCLC MMEarth-Bench Synthetic Acquisition budget T 10 7 25 10 5 varies Step threshold t0t_0 T/2T/2 T/2T/2 T/2T/2 T/2T/2 T/2T/2 T/2T/2 Gain factor λ 1.5 1.5 1.5 1.5 1.5 1.5 Discount factor γ 0.98 0.98 0.99 0.98 0.98 0.98 Training steps 300k 300k 500k 300k 200k 300k The step threshold t0t_0 is uniformly set to half of the episode length T. Across experiments, the acquisition budget T is chosen to lie between 5% and 25% of the total number of available features, depending on the dataset. We set γ=0.98γ=0.98 for small-horizon acquisition and γ=0.99γ=0.99 for long-horizon (hPancreas). For datasets with fewer features, we train for 300,000 steps; for larger feature spaces, we train for 500,000 steps (except for MMEarth-Bench, where we train for 200,000 steps). Appendix H Additional Experiments Figure 6: Downstream performance of random, prior-based ([j]E[ E_j]), optimal-test-time, and oracle policies on a random isotropic regression task H.1 Controlled Synthetic Experiment We first isolate the core question in a controlled setting with known data-generating process: how much does conditioning on partial observations help? For the synthetic data, we consider a randomized isotropic linear regression task with ∼(0,1dd) W (0, 1d I_d) and ε′∼Gumbel(μ′,σ′) (μ ,σ ) yielding labels y=⊤+ε′y= W x+ . We fit W via ridge regression on the full feature set (see Assumption 3.2), with labels from isotropic linear regression (details in § G.1.6). Fig. 6 confirms that the optimal test-time policy consistently outperforms both non-oracle baselines and closely tracks the oracle in early stages, validating the theoretical separation of §3 and motivating the long-horizon RL strategy of echo-k echo -k. H.2 Extended Baseline Comparisons In vision and tabular settings, features are individually cheap but collectively redundant; a policy that identifies the most informative subset early reduces measurement overhead at test time. For MNIST, Fashion-MNIST, and MiniBooNE, we train a masked autoencoder and freeze the encoder as echo-k’s backbone. For hPancreas, we use TABULA (Ding et al., 2025) scRNA foundation model. echo-k echo -k interfaces with it only through (⋅) z(·) and ⋆ z , requiring no architecture-specific changes (see § G.2 for backbone details). Results on MNIST and Fashion-MNIST. Both datasets are downsampled to 14×1414× 14 following Li and Oliva (2021), yielding d=196d=196 pixel-features with 10-class classification as the downstream task. The goal is to classify each image (handwritten digit or clothing item) by sequentially revealing as few pixels as possible. All methods start from the same top-corner pixel (Fig. 8). echo-k echo -k rapidly concentrates its acquisitions on discriminative image regions, achieving gains of up to 12% over all non-oracle baselines in the early regime (10–20% of pixels). No single baseline is consistently second-best: Variance leads on MNIST while Random is surprisingly competitive on Fashion-MNIST. Results on MiniBooNE. MiniBooNE contains d=33d=33 reconstructed physics variables from a neutrino experiment, with binary signal-vs-background classification as the downstream task; these variables are costly to compute and highly correlated (see § G.1). echo-k echo -k outperforms all heuristic baselines and the one-step oracle (Fig. 7). We attribute this to the oracle’s myopic strategy, which over-selects correlated variables early and suffers diminishing returns, whereas echo-k echo -k optimizes over a long horizon and learns to exploit complementary information. Figure 7: Results of echo-k echo -k across all baselines for the MNIST (top-left), Fashion-MNIST (top-right), hPancreas (bottom-left), and MiniBooNE (bottom-right) datasets. H.3 Comparison with Adaptive Acquisition Policies Most baselines in the main text are fixed acquisition orderings. To verify that the gains of echo-k echo -k are not solely due to comparison against fixed rankings, we compare against label-free controls that condition acquisition on the current instance and match the information setting of echo-k echo -k: • GSMRL-AIR, an instance-adaptive RL method that conditions action selection on the per-sample conditional imputations of a generative surrogate (Li and Oliva, 2021), adapted to the label-free setting. • Adaptive Latent Gain (ALG), a learned regressor that predicts the conditional one-step representation improvement directly from the current partial observation. ALG shares the reward and state representation of echo-k echo -k, but not its long-horizon objective. • Cluster Information Gain (CIG), which scores candidate views by mutual information under a fitted Gaussian mixture model posterior (MiniBooNE only). All methods observe only the currently acquired values and the acquisition mask, and never access the full-view embedding or downstream labels at deployment. Table 8: Comparison with adaptive acquisition policies. Accuracy (%) at different acquisition budgets on MiniBooNE and hPancreas. All policies operate without downstream labels. denotes best and second best performance. Method MiniBooNE Method hPancreas 20% 40% 60% 80% 20% 40% 60% 80% CIG 79.8079.80 84.8784.87 86.1786.17 86.9786.97 ALG 77.5077.50 83.6083.60 86.1286.12 87.2687.26 ALG 73.3873.38 80.1680.16 83.1283.12 86.3386.33 GSMRL-AIR 79.1879.18 84.7884.78 86.4086.40 86.2686.26 GSMRL-AIR 77.2377.23 81.8781.87 84.4184.41 86.4086.40 echo-k echo -k 81.7081.70 86.8786.87 87.6087.60 87.8387.83 echo-k echo -k 80.8180.81 88.0488.04 88.4388.43 88.4588.45 Table 8 shows that echo-k echo -k outperforms all adaptive controls at every budget on both datasets, including GSMRL-AIR, a fully adaptive RL baseline operating under the same label-free constraints. The comparison with ALG is particularly informative: both methods use representation improvement to guide acquisition, but echo-k echo -k optimizes gains over a longer horizon. This indicates that the advantage of echo-k echo -k comes from long-horizon planning rather than from the recovery signal alone. H.4 Comparison with Supervised Acquisition Table 9 reports the full comparison of echo-k echo -k with five supervised acquisition methods on MiniBooNE: DIME (Gadgil et al., 2023), GDFS (Covert et al., 2023), SEFA (Norcliffe et al., 2025), GSMRL (Li and Oliva, 2021), and EDDI (Ma et al., 2019). Unlike echo-k echo -k, these methods use downstream labels during policy learning. Table 9: Comparison with supervised acquisition on MiniBooNE. Accuracy (%) at different acquisition budgets. All methods except echo-k echo -k use downstream labels. denotes best and second best performance. Method 10% 20% 40% 60% DIME 79.9779.97 84.2384.23 86.8786.87 87.0787.07 GDFS 80.2380.23 84.7384.73 87.5087.50 87.8087.80 SEFA 78.6778.67 84.7084.70 87.7387.73 88.1388.13 GSMRL 75.7375.73 83.2783.27 87.3787.37 87.7787.77 EDDI 72.3072.30 76.7076.70 83.5083.50 85.0785.07 echo-k echo -k 80.0380.03 81.7081.70 86.8786.87 87.6087.60 At 10% acquisition, echo-k echo -k is within 0.2 points of the best supervised method and exceeds four of the five, despite never observing downstream labels. The gap is largest at 20%, where GDFS leads by 3.0 points, and narrows to 0.9 and 0.5 points at 40% and 60%, respectively. Given that echo-k echo -k never observes downstream labels during training, we view this as the expected cost of task agnosticism. H.5 Cost of Task Agnosticism When a single downstream task is fixed and sufficient labels are available, a task-specific policy may be preferable. To quantify this trade-off, we train supervised acquisition policies for NSCLC subtyping (S) and grading (G) using the same architecture and training budget as echo-k echo -k, replacing the latent reward with the downstream-loss improvement rtsup=ℓ(g((t−1)),y)−ℓ(g((t)),y).r_t^sup= (g( z( P_t-1)),y )- (g( z( P_t)),y ). PPO-Subtype and PPO-Grade optimize their respective endpoints, while PPO-Multitask averages the two task-specific losses at each step. We evaluate each policy on both tasks. Table 10: Task-specific and task-agnostic acquisition on NSCLC. Accuracy (%) for subtyping (S), grading (G), and their mean at different acquisition budgets. Supervised PPO policies use task labels during policy learning, while echo-k echo -k uses only the representation-recovery reward. Policy 20% 40% 60% S G Mean S G Mean S G Mean PPO-Subtype 83.283.2 62.862.8 73.073.0 85.485.4 64.264.2 74.874.8 86.386.3 65.065.0 75.775.7 PPO-Grade 78.578.5 67.867.8 73.173.1 82.282.2 68.268.2 75.275.2 83.183.1 68.068.0 75.575.5 PPO-Multitask 82.982.9 66.566.5 74.774.7 84.184.1 67.267.2 75.675.6 84.884.8 67.967.9 76.376.3 echo-k echo -k 82.882.8 65.865.8 74.374.3 83.883.8 67.167.1 75.575.5 85.285.2 67.467.4 76.376.3 At 20% acquisition, echo-k echo -k trails the corresponding task-specific policies by 0.4 points on subtyping and 2.0 points on grading (Table 10). The specialized policies also transfer poorly across endpoints, losing approximately 3–5 points when evaluated on the other task. More importantly, echo-k echo -k is within 0.4, 0.1, and 0.0 points of PPO-Multitask in mean accuracy at 20%, 40%, and 60%, respectively, despite using neither task label. This quantifies the expected cost of task agnosticism, while showing that preserving the pretrained representation nearly matches direct multitask supervision. H.6 Robustness to Partially Observed Data Table 11 provides additional robustness results under partially observed reference data on MNIST-14 and FMNIST-14. We train echo-k echo -k after randomly masking an m-fraction of views when computing the reference representation used for the self-supervised reward. Across both datasets, performance degrades smoothly as m increases, rather than collapsing. On MNIST-14, partially observed-reference policies remain close to or even slightly above the complete-reference policy across several budgets, indicating that the acquisition policy can tolerate substantial noise in the reference target. On FMNIST-14, the degradation is more visible but remains gradual: even with 50% missing reference views, echo-k echo -k stays competitive with strong baselines and preserves most of its early-budget advantage. These results complement the biological experiments in Table 1, showing that the method’s robustness to incomplete reference data is not specific to a single modality or dataset. H.7 Heterogeneous Acquisition Costs To test whether echo-k echo -k can adapt when views have nonuniform acquisition costs, we run a controlled cost-aware experiment on MNIST-14. We assign each pixel a sample-dependent acquisition cost based on its intensity, rescaled to the range [0.25,2.0][0.25,2.0], so that the policy must trade off representation recovery against cumulative cost. We compare a cost-aware variant of echo-k echo -k, trained with a cost-penalized reward and cost information in the state, against the standard cost-unaware policy. Fig. 8 shows that cost-aware training changes the acquisition behavior. Over the first ten acquisitions, the cost-aware policy selects substantially cheaper pixels on average than the cost-unaware policy, reducing the mean acquisition cost from 0.89 to 0.56. This shift also improves performance in the low-budget regime: when accuracy is plotted against cumulative cost rather than number of pixels, the cost-aware policy reaches high accuracy with fewer cost units. As the budget grows, both policies converge, indicating that cost-awareness is most useful when acquisition resources are limited. This controlled experiment confirms that echo-k echo -k can incorporate heterogeneous costs by changing the reward and state, without modifying the representation-recovery objective. Figure 8: a. Overview of the task and acquisition behavior of echo-k: we illustrate how echo-k echo -k sequentially acquires pixels in an unsupervised manner, progressively concentrating the (unobserved) downstream class probability mass. b. Distribution of acquisition costs among the first ten selected pixels; the cost-aware policy selects lower-cost pixels early. c. Accuracy as a function of cumulative acquisition cost; cost-aware training improves performance in the low-budget regime, while both policies converge at larger budgets. Table 11: We train echo-k echo -k after randomly masking an m-fraction of views when computing the reference representation. Accuracy is reported at different acquisition budgets on MNIST-14 and FMNIST-14. denotes best and second best performance, apart from the Oracle. Method MNIST-14 FMNIST-14 20% 40% 60% 80% 20% 40% 60% 80% Oracle 0.937±0.0050.937 ± 0.005 0.951±0.0020.951 ± 0.002 0.953±0.0020.953 ± 0.002 0.956±0.0020.956 ± 0.002 0.824±0.0030.824 ± 0.003 0.843±0.0060.843 ± 0.006 0.840±0.0030.840 ± 0.003 0.842±0.0040.842 ± 0.004 Variance 0.759±0.0070.759 ± 0.007 0.865±0.0050.865 ± 0.005 0.928±0.0050.928 ± 0.005 0.951±0.0060.951 ± 0.006 0.565±0.0220.565 ± 0.022 0.670±0.0080.670 ± 0.008 0.725±0.0160.725 ± 0.016 0.741±0.0050.741 ± 0.005 spec-φ3 _3 0.168±0.0140.168 ± 0.014 0.754±0.0180.754 ± 0.018 0.894±0.0030.894 ± 0.003 0.942±0.0020.942 ± 0.002 0.543±0.0360.543 ± 0.036 0.600±0.0390.600 ± 0.039 0.681±0.0120.681 ± 0.012 0.738±0.0130.738 ± 0.013 MCFS 0.678±0.0100.678 ± 0.010 0.829±0.0050.829 ± 0.005 0.904±0.0040.904 ± 0.004 0.918±0.0020.918 ± 0.002 0.631±0.0290.631 ± 0.029 0.760±0.0160.760 ± 0.016 0.767±0.0130.767 ± 0.013 0.793±0.0100.793 ± 0.010 Random 0.627±0.0090.627 ± 0.009 0.800±0.0050.800 ± 0.005 0.871±0.0030.871 ± 0.003 0.907±0.0030.907 ± 0.003 0.708±0.0040.708 ± 0.004 0.779±0.0040.779 ± 0.004 0.805±0.0020.805 ± 0.002 0.821±0.0050.821 ± 0.005 echo-k echo -k 0.830±0.0040.830 ± 0.004 0.913±0.0120.913 ± 0.012 0.935±0.0090.935 ± 0.009 0.951±0.0030.951 ± 0.003 0.749±0.0190.749 ± 0.019 0.793±0.0070.793 ± 0.007 0.818±0.0070.818 ± 0.007 0.833±0.0090.833 ± 0.009 echo-k echo -k (m=m=10%) 0.845±0.0150.845 ± 0.015 0.921±0.0020.921 ± 0.002 0.941±0.0020.941 ± 0.002 0.949±0.0030.949 ± 0.003 0.742±0.0240.742 ± 0.024 0.780±0.0080.780 ± 0.008 0.814±0.0080.814 ± 0.008 0.821±0.0110.821 ± 0.011 echo-k echo -k (m=m=20%) 0.862±0.0020.862 ± 0.002 0.913±0.0060.913 ± 0.006 0.939±0.0070.939 ± 0.007 0.948±0.0030.948 ± 0.003 0.735±0.0200.735 ± 0.020 0.767±0.0130.767 ± 0.013 0.803±0.0060.803 ± 0.006 0.813±0.0080.813 ± 0.008 echo-k echo -k (m=m=30%) 0.836±0.0080.836 ± 0.008 0.912±0.0030.912 ± 0.003 0.940±0.0080.940 ± 0.008 0.950±0.0050.950 ± 0.005 0.721±0.0180.721 ± 0.018 0.769±0.0140.769 ± 0.014 0.798±0.0090.798 ± 0.009 0.814±0.0080.814 ± 0.008 echo-k echo -k (m=m=40%) 0.830±0.0170.830 ± 0.017 0.903±0.0140.903 ± 0.014 0.934±0.0010.934 ± 0.001 0.950±0.0110.950 ± 0.011 0.716±0.0240.716 ± 0.024 0.776±0.0030.776 ± 0.003 0.787±0.0030.787 ± 0.003 0.801±0.0090.801 ± 0.009 echo-k echo -k (m=m=50%) 0.823±0.0190.823 ± 0.019 0.900±0.0110.900 ± 0.011 0.938±0.0070.938 ± 0.007 0.949±0.0020.949 ± 0.002 0.711±0.0130.711 ± 0.013 0.765±0.0180.765 ± 0.018 0.792±0.0060.792 ± 0.006 0.815±0.0020.815 ± 0.002 H.8 Adaptive Stopping A fixed acquisition horizon may be restrictive in deployment, where the policy should also decide whether the currently acquired views are already sufficient. We extend echo-k echo -k with a STOP action, converting the fixed-horizon MDP into an optimal-stopping problem. For a measurement action a, the reward is rt=e(t−1)−e(t)e(0)+ϵ−λca,e()=‖()−⋆‖2,r_t= e( P_t-1)-e( P_t)e( P_0)+ε-λ c_a, e( P)=\| z( P)- z \|_2, while STOP terminates the episode with zero additional reward. Since STOP yields zero future reward, the policy continues only when the expected cumulative representation gain outweighs the acquisition cost, and with γ=1γ=1, the learned stopping time τ directly trades recovery against total acquisition cost. As in echo-k echo -k, ⋆ z is used only to construct the label-free training reward; at deployment, both acquisition and stopping depend only on the current partial state. Table 12: Adaptive stopping on MiniBooNE. Accuracy (%) and number of acquired features. Policy # Features Accuracy echo-k echo -k (fixed horizon) 1010 86.1186.11 echo-k echo -k (STOP) 9.869.86 (mean) 86.0686.06 On MiniBooNE, echo-k echo -k with stopping reaches 86.06% accuracy at a mean of 9.86 acquired features, showing instance-dependent stopping (Table 12). This matches the fixed-horizon policy at a comparable budget (86.11% at t=10t=10), while requiring no budget to be specified in advance. Together with the heterogeneous-cost experiment above, this shows that the representation-recovery objective of echo-k echo -k can be combined with cost-aware rewards to decide both which views to acquire and when to stop. H.9 Empirical Validation of the Near Isometry Assumption A key ingredient in our theoretical analysis is Assumption 3.4 (Near Isometry), which requires that the normalized columns of the embedding weight matrix are approximately orthogonal. We validate this assumption empirically on two classes of models. H.9.1 DinoV2 Foundation Models We examine the DINOv2 family of vision transformers (Oquab et al., 2023), including ViT-S/14, ViT-B/14, ViT-L/14, and ViT-G/14 (with register tokens). For each model, we: (1) extract the final MLP projection weights ∈ℝp×d W ^p× d from the last transformer block’s feed-forward network (fc2 layer); (2) normalize columns to get B and compute the Gram matrix =⊤ G= B B; (3) analyze the distribution of the off-diagonal elements of −d G- I_d. As shown in Fig. 9, these values concentrate tightly near zero across all model scales, confirming approximate orthogonality. Larger models exhibit tighter concentration, consistent with theoretical expectations for high-dimensional random-like weight matrices. Figure 9: Distribution of ⊤−d B B- I_d for 4 different DinoV2 models of increasing capacity. H.9.2 Trained Autoencoders We further validate the assumption on trained autoencoders across varying dimensions d from 32 to 4096. For each dimension, we generate low-rank data ⊤=⊤+ϵ X = U V + ε where ∈ℝn×r U ^n× r, ∈ℝd×r V ^d× r. We extract the last layer weight matrix W and compute the effective rank peffp_eff using the entropy of normalized singular values (Roy and Vetterli, 2007): peff=exp(−∑isi∑ksklogsi∑ksk).p_eff= (- _i s_i _ks_k s_i _ks_k ). We sample random subsets J⊆1,…,dJ \1,…,d\ of varying sizes m, normalize the corresponding weight vectors to form J B_J, and compute ‖J⊤J−m‖op\| B_J B_J- I_m\|_op. We compare against the theoretical bound C(m/peff+m/peff)C ( m/p_eff+m/p_eff ). Across 50 repetitions, the near isometry property holds with C∈[0.93,1.73]C∈[0.93,1.73] across all tested dimensions. As d increases, the standard deviation of off-diagonal elements decreases, validating that trained neural network weights naturally form near-orthonormal systems (Fig. 10). Figure 10: Variation of ‖⊤−d‖op\| B B- I_d\|_op for varying values of d on trained autoencoder models and variation of the empirical estimation of C with d. Figure 11: Analysis of the top-15 markers and their frequency, as chosen by echo-k echo -k jointly for the cancer subtyping and grading task on the NSCLC spatial proteomics dataset. H.10 Analysis of Markers Acquired by echo-k echo -k on Spatial Proteomics In Fig. 11, we analyze the distribution of markers chosen by echo-k echo -k on the NSCLC dataset tasks: cancer subtyping and grading from the marker panel provided by (Cords et al., 2024). In an unsupervised fashion, echo-k echo -k prioritizes markers that are well-established in the clinical literature for their roles in tumor characterization and microenvironmental context: • panCK: a pan-cytokeratin marker for epithelial (tumor) cell identity, consistently selected first. • Ki-67: a proliferation marker widely used in clinical grading of lung cancers (Martin et al., 2004). • CD4: a marker of helper T-cell infiltration, reflecting the adaptive immune response in the tumor microenvironment. • LYVE-1: a marker of lymphatic endothelial cells, relevant to lymphatic involvement and metastatic potential. This early selection pattern suggests that echo-k echo -k effectively identifies markers with strong relevance to both tumor biology and immune context, thereby supporting both subtyping and grading tasks without any supervision. 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